Chapter 5

Series

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Section 5.1Numerical series, tests for convergence of series of constant sign[123/89%]
Problem 2546

Directly prove the convergence of the following series and find their sums:

1−12+14−18+⋯+(−1)n−12n−1+⋯ .1-\dfrac {1}{2}+\dfrac {1}{4}-\dfrac {1}{8}+\cdots +\dfrac {(-1)^{n-1}}{2^{n-1}}+\cdots .

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Problem 2547

Directly prove the convergence of the following series and find their sums: (12+13)+(122+132)+⋯+(12n+13n)+⋯ .\left(\dfrac {1}{2}+\dfrac {1}{3}\right)+\left(\dfrac {1}{2^{2}}+\dfrac {1}{3^{2}}\right)+\cdots +\left(\dfrac {1}{2^{n}}+\dfrac {1}{3^{n}}\right)+\cdots .

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Problem 2548

Directly prove the convergence of the following series and find their sums: 12+322+523+⋯+2n−12n+⋯ .\dfrac {1}{2}+\dfrac {3}{2^{2}}+\dfrac {5}{2^{3}}+\cdots +\dfrac {2n-1}{2^{n}}+\cdots .

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Problem 2549

Directly prove the convergence of the following series and find their sums: 11⋅2+12⋅3+13⋅4+⋯+1n(n+1)+⋯ .\dfrac {1}{1 \cdot 2}+\dfrac {1}{2 \cdot 3}+\dfrac {1}{3 \cdot 4}+\cdots +\dfrac {1}{n(n+1)}+\cdots .

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Problem 2550

Directly prove the convergence of the following series and find their sums: 11⋅4+14⋅7+⋯+1(3n−2)(3n+1)+⋯ .\dfrac {1}{1 \cdot 4}+\dfrac {1}{4 \cdot 7}+\cdots +\dfrac {1}{(3n-2)(3n+1)}+\cdots .

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Problem 2551

Directly prove the convergence of the following series and find their sums:

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(a)

qsin⁡α+q2sin⁡2α+⋯+qnsin⁡nα+⋯(∣q∣<1)q \sin \alpha +q^{2} \sin 2\alpha +\cdots +q^{n} \sin n\alpha +\cdots \quad (\left|q\right|<1);

(b)

qcos⁡α+q2cos⁡2α+⋯+qncos⁡nα+⋯(∣q∣<1)q \cos \alpha +q^{2} \cos 2\alpha +\cdots +q^{n} \cos n\alpha +\cdots \quad (\left|q\right|<1).

Problem 2552

Directly prove the convergence of the following series and find its sum:∑n=1∞(n−2−2n+1+n)\displaystyle \sum_{n=1}^{\infty }\left(\sqrt{n-2}-2\sqrt{n+1}+\sqrt{n}\right).

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Problem 2553

Investigate the convergence of the series ∑n=1∞sin⁡nx\displaystyle \sum_{n=1}^{\infty } \sin nx.

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Problem 2554

Prove that if the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges, then the series obtained by grouping the terms of this series without changing their order

∑n=1∞An,where An=∑i=pnpn+1−1ai(p1=1,p1<p2<⋯ ) \sum _{n=1}^{\infty } A_{n}, \quad \text{where } A_{n}=\sum _{i=p_{n}}^{p_{n+1}-1} a_{i} \quad (p_{1}=1, p_{1}<p_{2}<\cdots )

also converges and has the same sum. The converse is not true. Give an example.

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Problem 2555

Prove that if the terms of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} are positive, and the series ∑n=1∞An\displaystyle \sum_{n=1}^{\infty } A_{n} obtained by grouping the terms of this series converges, then the original series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} also converges.

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Problem 2556

Investigate the convergence of the following series: 1−1+1−1+1−1+⋯ .1-1+1-1+1-1+\cdots .

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Problem 2557

Investigate the convergence of the following series: 0.001+0.001+0.0013+⋯ .0.001+\sqrt{0.001}+\sqrt[3]{0.001}+\cdots .

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Problem 2558

Investigate the convergence of the following series: 11!+12!+13!+⋯+1n!+⋯ .\dfrac {1}{1!}+\dfrac {1}{2!}+\dfrac {1}{3!}+\cdots +\dfrac {1}{n!}+\cdots .

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Problem 2559

Investigate the convergence of the following series: 1+13+15+17+⋯+12n−1+⋯ .1+\dfrac {1}{3}+\dfrac {1}{5}+\dfrac {1}{7}+\cdots +\dfrac {1}{2n-1}+\cdots .

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Problem 2560

Investigate the convergence of the following series: 11001+12001+13001+⋯+11000n+1+⋯ .\dfrac {1}{1001}+\dfrac {1}{2001}+\dfrac {1}{3001}+\cdots +\dfrac {1}{1000n+1}+\cdots .

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Problem 2561

Investigate the convergence of the following series: 1+23+35+⋯+n2n−1+⋯ .1+\dfrac {2}{3}+\dfrac {3}{5}+\cdots +\dfrac {n}{2n-1}+\cdots .

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Problem 2562

Investigate the convergence of the following series: 1+132+152+⋯+1(2n−1)2+⋯ .1+\dfrac {1}{3^{2}}+\dfrac {1}{5^{2}}+\cdots +\dfrac {1}{(2n-1)^{2}}+\cdots .

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Problem 2563

Investigate the convergence of the following series: 12+123+134+⋯+1nn+1+⋯ .\dfrac {1}{\sqrt{2}}+\dfrac {1}{2\sqrt{3}}+\dfrac {1}{3\sqrt{4}}+\cdots +\dfrac {1}{n\sqrt{n+1}}+\cdots .

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Problem 2564

Investigate the convergence of the following series: 11⋅3+13⋅5+⋯+1(2n−1)(2n+1)+⋯ .\dfrac {1}{\sqrt{1 \cdot 3}}+\dfrac {1}{\sqrt{3 \cdot 5}}+\cdots +\dfrac {1}{\sqrt{(2n-1)(2n+1)}}+\cdots .

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Problem 2565

Prove that the series formed from the reciprocals of the terms of an arithmetic series is divergent.

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Problem 2566

Prove that if the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} (AA) and ∑n=1∞bn\displaystyle \sum_{n=1}^{\infty } b_{n} (BB) both converge and an≤cn≤bna_{n} \leq c_{n} \leq b_{n} (n=1,2,…)(n=1,2,\ldots ), then the series ∑n=1∞cn\displaystyle \sum_{n=1}^{\infty } c_{n} (CC) also converges. If the series (AA) and (BB) both diverge, what can be said about the convergence of series (CC)?

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Problem 2567

Suppose the two divergent series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} and ∑n=1∞bn\displaystyle \sum_{n=1}^{\infty } b_{n} have terms that are nonnegative. What can be said about the convergence of the following two series:

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(a)

∑n=1∞min⁡(an,bn)\displaystyle \sum_{n=1}^{\infty } \min (a_{n}, b_{n}) and

(b)

∑n=1∞max⁡(an,bn)\displaystyle \sum_{n=1}^{\infty } \max (a_{n}, b_{n})?

Problem 2568

Prove that if the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} (an≥0)(a_{n} \geq 0) converges, then the series ∑n=1∞an2\displaystyle \sum_{n=1}^{\infty } a_{n}^{2} also converges. The converse does not hold; give an example.

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Problem 2569

Prove that if the series ∑n=1∞an2\displaystyle \sum_{n=1}^{\infty } a_{n}^{2} and ∑n=1∞bn2\displaystyle \sum_{n=1}^{\infty } b_{n}^{2} converge, then the series

∑n=1∞∣anbn∣,∑n=1∞(an+bn)2,∑n=1∞∣an∣n \sum _{n=1}^{\infty } \left|a_{n} b_{n}\right|, \quad \sum _{n=1}^{\infty }(a_{n}+b_{n})^{2}, \quad \sum _{n=1}^{\infty } \frac{\left|a_{n}\right|}{n}

also converge.

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Problem 2570

Prove that if

lim⁡n→∞nan=a≠0, \lim _{n \rightarrow \infty } na_{n}=a \neq 0,

then the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} diverges.

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Problem 2571

Prove that if the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, whose terms are positive and monotonically decreasing, converges, then

lim⁡n→∞nan=0. \lim _{n \rightarrow \infty } na_{n}=0 .
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Problem 2572

If for p=1,2,3,…p=1,2,3,\ldots, lim⁡n→∞(an+1+an+2+⋯+an+p)=0\displaystyle \lim_{n \rightarrow \infty }(a_{n+1}+a_{n+2}+\cdots +a_{n+p})=0. Does the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converge?

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Problem 2573

Using the Cauchy criterion, prove the convergence of the following series of positive terms: a0+a110+⋯+an10n+⋯(∣an∣<10)a_{0}+\dfrac {a_{1}}{10}+\cdots +\dfrac {a_{n}}{10^{n}}+\cdots \quad (\left|a_{n}\right|<10).

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Problem 2574

Using the Cauchy criterion, prove the convergence of the following series of positive terms: sin⁡x2+sin⁡2x22+⋯+sin⁡nx2n+⋯ .\dfrac {\sin x}{2}+\dfrac {\sin 2x}{2^{2}}+\cdots +\dfrac {\sin nx}{2^{n}}+\cdots .

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Problem 2575

Using the Cauchy criterion, prove the convergence of the following series of positive terms: cos⁡x−cos⁡2x1+cos⁡2x−cos⁡3x2+⋯+cos⁡nx−cos⁡(n+1)xn+⋯ .\dfrac {\cos x-\cos 2x}{1}+\dfrac {\cos 2x-\cos 3x}{2}+\cdots +\dfrac {\cos nx-\cos (n+1)x}{n}+\cdots .

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Problem 2575.1
[NO TASK]
Problem 2575.1
[NO TASK]
Problem 2576

Using the Cauchy criterion, prove the divergence of the following series: 1+12+13+⋯+1n+⋯ .1+\dfrac {1}{2}+\dfrac {1}{3}+\cdots +\dfrac {1}{n}+\cdots .

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Problem 2577

Using the Cauchy criterion, prove the divergence of the following series: 1+12−13+14+15−16+⋯ .1+\dfrac {1}{2}-\dfrac {1}{3}+\dfrac {1}{4}+\dfrac {1}{5}-\dfrac {1}{6}+\cdots .

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Problem 2577.1
[NO TASK]
Problem 2577.1
[NO TASK]
Problem 2578

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series: 10001!+100022!+100033!+⋯+1000nn!+⋯ .\dfrac {1000}{1!}+\dfrac {1000^{2}}{2!}+\dfrac {1000^{3}}{3!}+\cdots +\dfrac {1000^{n}}{n!}+\cdots .

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Problem 2579

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series: (1!)22!+(2!)24!+⋯+(n!)2(2n)!+⋯ .\dfrac {(1!)^{2}}{2!}+\dfrac {(2!)^{2}}{4!}+\cdots +\dfrac {(n!)^{2}}{(2n)!}+\cdots .

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Problem 2580

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series: 1!1+2!22+3!33+⋯+n!nn+⋯ .\dfrac {1!}{1}+\dfrac {2!}{2^{2}}+\dfrac {3!}{3^{3}}+\cdots +\dfrac {n!}{n^{n}}+\cdots .

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Problem 2581

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:

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(a)

2⋅1!11+22⋅2!22+23⋅3!33+⋯+2nn!nn+⋯\dfrac {2 \cdot 1!}{1^{1}}+\dfrac {2^{2} \cdot 2!}{2^{2}}+\dfrac {2^{3} \cdot 3!}{3^{3}}+\cdots +\dfrac {2^{n} n!}{n^{n}}+\cdots;

(b)

3⋅1!11+32⋅2!22+33⋅3!33+⋯+3nn!nn+⋯ .\dfrac {3 \cdot 1!}{1^{1}}+\dfrac {3^{2} \cdot 2!}{2^{2}}+\dfrac {3^{3} \cdot 3!}{3^{3}}+\cdots +\dfrac {3^{n} n!}{n^{n}}+\cdots .

Problem 2582

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:(1!)22+(2!)224+(3!)229+⋯+(n!)22n2+⋯ .\dfrac {(1!)^{2}}{2}+\dfrac {(2!)^{2}}{2^{4}}+\dfrac {(3!)^{2}}{2^{9}}+\cdots +\dfrac {(n!)^{2}}{2^{n^{2}}}+\cdots .

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Problem 2583

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:10001+1000⋅10011⋅3+1000⋅1001⋅10021⋅3⋅5+⋯ .\dfrac {1000}{1}+\dfrac {1000 \cdot 1001}{1 \cdot 3}+\dfrac {1000 \cdot 1001 \cdot 1002}{1 \cdot 3 \cdot 5}+\cdots .

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Problem 2584

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:42+4⋅72⋅6+4⋅7⋅102⋅6⋅10+⋯ .\dfrac {4}{2}+\dfrac {4 \cdot 7}{2 \cdot 6}+\dfrac {4 \cdot 7 \cdot 10}{2 \cdot 6 \cdot 10}+\cdots .

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Problem 2585

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:∑n=1∞(2−23)(2−25)⋯(2−22n+1)\displaystyle \sum_{n=1}^{\infty }\left(\sqrt{2}-\sqrt[3]{2}\right)\left(\sqrt{2}-\sqrt[5]{2}\right)\cdots \left(\sqrt{2}-\sqrt[2n+1]{2}\right).

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Problem 2585.1
[NO TASK]
Problem 2585.1
[NO TASK]
Problem 2585.2
[NO TASK]
Problem 2585.2
[NO TASK]
Problem 2586

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:∑n=1∞n2(2+1n)n\displaystyle \sum_{n=1}^{\infty } \frac{n^{2}}{\left(2+\frac{1}{n}\right)^{n}}.

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Problem 2587

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:∑n=1∞nn−1n(n+1n)n\displaystyle \sum_{n=1}^{\infty } \frac{n^{n-\frac{1}{n}}}{\left(n+\frac{1}{n}\right)^{n}}.

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Problem 2588

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:∑n=2∞1ln⁡nn\displaystyle \sum_{n=2}^{\infty } \frac{1}{\sqrt[n]{\ln n}}.

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Problem 2589

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:∑n=1∞nn−1(2n2+n+1)n+12\displaystyle \sum_{n=1}^{\infty } \frac{n^{n-1}}{(2n^{2}+n+1)^{\frac{n+1}{2}}}.

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Problem 2589.1
[NO TASK]
Problem 2589.1
[NO TASK]
Problem 2589.2
[NO TASK]
Problem 2589.2
[NO TASK]
Problem 2590

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:2+2−2+2−2+2+2−2+2+2+⋯ .\sqrt{2}+\sqrt{2-\sqrt{2}}+\sqrt{2-\sqrt{2+\sqrt{2}}}+\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}+\cdots .

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Problem 2591

Prove: if

lim⁡n→∞an+1an=q(an>0), \lim _{n \rightarrow \infty } \frac{a_{n+1}}{a_{n}}=q \quad (a_{n}>0),

then an=o(q1n)a_{n}=o(q_{1}^{n}), where q1>qq_{1}>q.

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Problem 2591.1
[NO TASK]
Problem 2591.1
[NO TASK]
Problem 2591.2
[NO TASK]
Problem 2591.2
[NO TASK]
Problem 2592

Prove: if

lim‾⁡n→∞an+1an=q<1(an>0), \varlimsup _{n \rightarrow \infty } \frac{a_{n+1}}{a_{n}}=q<1 \quad (a_{n}>0),

then the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges.

The converse statement is not true. Study the example

12+13+122+132+123+133+⋯ . \frac{1}{2}+\frac{1}{3}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{2^{3}}+\frac{1}{3^{3}}+\cdots .
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Problem 2593

Prove that if, for the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} (an>0)(a_{n}>0), the limit

lim⁡n→∞an+1an=q(A) \lim _{n \rightarrow \infty } \frac{a_{n+1}}{a_{n}}=q \tag {A}

exists, then

lim⁡n→∞ann=q(Б) \lim _{n \rightarrow \infty } \sqrt[n]{a_{n}}=q \tag {Б}

also exists.

The converse statement is not true: if limit (Б) exists, then limit (A) may fail to exist. Study the example

∑n=1∞3+(−1)n2n+1. \sum _{n=1}^{\infty } \frac{3+(-1)^{n}}{2^{n+1}} .
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Problem 2594

Prove that if

lim‾⁡n→∞ann=q(an≥0), \varlimsup _{n \rightarrow \infty } \sqrt[n]{a_{n}}=q \quad (a_{n} \geq 0),

then (a) when q<1q<1 the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges; (b) when q>1q>1 this series diverges (a generalization of Cauchy's test).

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Problem 2595

Investigate the convergence of the following series: ∑n=1∞2+(−1)n2n\displaystyle \sum_{n=1}^{\infty } \frac{2+(-1)^{n}}{2^{n}}.

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Problem 2596

Investigate the convergence of the following series: ∑n=1∞ncos⁡2nπ32n\displaystyle \sum_{n=1}^{\infty } \frac{n\cos^{2}\dfrac {n\pi }{3}}{2^{n}}.

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Problem 2597

Investigate the convergence of the following series: ∑n=1∞n3[2+(−1)n]n3n\displaystyle \sum_{n=1}^{\infty } \frac{n^{3}\left[\sqrt{2}+(-1)^{n}\right]^{n}}{3^{n}}.

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Problem 2597.1
[NO TASK]
Problem 2597.1
[NO TASK]
Problem 2598

Using Raabe's and Gauss's tests, investigate the convergence of the following series: (12)p+(1⋅32⋅4)p+(1⋅3⋅52⋅4⋅6)p+⋯ .\left(\dfrac {1}{2}\right)^{p}+\left(\dfrac {1 \cdot 3}{2 \cdot 4}\right)^{p}+\left(\dfrac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^{p}+\cdots .

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Problem 2599

Using Raabe's and Gauss's tests, investigate the convergence of the following series: ab+a(a+d)b(b+d)+a(a+d)(a+2d)b(b+d)(b+2d)+⋯(a>0,b>0,d>0)\dfrac {a}{b}+\dfrac {a(a+d)}{b(b+d)}+\dfrac {a(a+d)(a+2d)}{b(b+d)(b+2d)}+\cdots \quad (a>0, b>0, d>0).

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Problem 2600

Using Raabe's and Gauss's tests, investigate the convergence of the following series: ∑n=1∞n!ennn+p\displaystyle \sum_{n=1}^{\infty } \frac{n! e^{n}}{n^{n+p}}.

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Problem 2601

Using Raabe's and Gauss's tests, investigate the convergence of the following series: ∑n=1∞n!(2+1)(2+2)⋯(2+n)\displaystyle \sum_{n=1}^{\infty } \frac{\sqrt{n!}}{(2+\sqrt{1})(2+\sqrt{2})\cdots (2+\sqrt{n})}.

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Problem 2602

Using Raabe's and Gauss's tests, investigate the convergence of the following series: ∑n=1∞n!n−pq(q+1)⋯(q+n)(p>0,q>0)\displaystyle \sum_{n=1}^{\infty } \frac{n! n^{-p}}{q(q+1)\cdots (q+n)} \quad (p>0, q>0).

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Problem 2603

Using Raabe's and Gauss's tests, investigate the convergence of the following series: ∑n=1∞p(p+1)⋯(p+n−1)n!⋅1nq(p>0,q>0)\displaystyle \sum_{n=1}^{\infty } \frac{p(p+1)\cdots (p+n-1)}{n!} \cdot \frac{1}{n^{q}} \quad (p>0, q>0).

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Problem 2604

Using Raabe's and Gauss's tests, investigate the convergence of the following series: ∑n=1∞(1⋅3⋅5⋯(2n−1)2⋅4⋅6⋯(2n))p⋅1nq\displaystyle \sum_{n=1}^{\infty }\left(\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}\right)^{p} \cdot \frac{1}{n^{q}}.

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Problem 2605

Using Raabe's and Gauss's tests, investigate the convergence of the following series: ∑n=1∞1np(1−xln⁡nn)n(p>0)\displaystyle \sum_{n=1}^{\infty } \frac{1}{n^{p}}\left(1-\frac{x\ln n}{n}\right)^{n} \quad (p>0).

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Problem 2606

Prove that if an>0a_{n}>0 (n=1,2,…)(n=1,2,\ldots ) and

lim⁡n→∞n(anan+1−1)=p, \lim _{n \rightarrow \infty } n\left(\frac{a_{n}}{a_{n+1}}-1\right)=p,

then an=o(1np−ε)a_{n}=o\left(\dfrac {1}{n^{p-\varepsilon }}\right) (ε>0)(\varepsilon >0).

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Problem 2607

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=np+a1np−1+⋯+apnq+b1nq−1+⋯+bqa_{n}=\dfrac {n^{p}+a_{1}n^{p-1}+\cdots +a_{p}}{n^{q}+b_{1}n^{q-1}+\cdots +b_{q}}, and nq+b1nq−1+⋯+bq>0n^{q}+b_{1}n^{q-1}+\cdots +b_{q}>0.

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Problem 2608

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=1npsin⁡πna_{n}=\dfrac {1}{n^{p}}\sin \dfrac {\pi }{n}.

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Problem 2609

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=(n+1−n)pln⁡n−1n+1a_{n}=\left(\sqrt{n+1}-\sqrt{n}\right)^{p}\ln \dfrac {n-1}{n+1} (n>1)(n>1).

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Problem 2610

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=ln⁡p(sec⁡πn)a_{n}=\ln^{p}\left(\sec \dfrac {\pi }{n}\right).

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Problem 2611

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=lg⁡bn(1+an)a_{n}=\lg_{b^{n}}\left(1+\sqrt[n]{a}\right) (a>0,b>0)(a>0, b>0).

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Problem 2612

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=[e−(1+1n)n]pa_{n}=\left[e-\left(1+\dfrac {1}{n}\right)^{n}\right]^{p}.

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Problem 2613

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=1n1+kln⁡na_{n}=\dfrac {1}{n^{1+\frac{k}{\ln n}}}.

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Problem 2614

Find the order of decrease of the general term ana_{n}, and hence investigate the convergence of the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}, where: an=1n1+1na_{n}=\dfrac {1}{n^{1+\frac{1}{n}}}.

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Problem 2614.1
[NO TASK]
Problem 2614.1
[NO TASK]
Problem 2615

Prove that if there exists α>0\alpha >0 such that for n≥n0n \geq n_{0}, ln⁡1anln⁡n≥1+α\dfrac {\ln \dfrac {1}{a_{n}}}{\ln n} \geq 1+\alpha (an>0)(a_{n}>0), then the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges; if from some n0n_{0} on, ln⁡1anln⁡n≤1\dfrac {\ln \dfrac {1}{a_{n}}}{\ln n} \leq 1, then the series diverges.

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Problem 2616

Investigate the convergence of the series with the following general term: an=nln⁡xa_{n}=n^{\ln x} (x>0)(x>0).

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Problem 2617

Investigate the convergence of the series with the following general term: an=1(ln⁡ln⁡n)ln⁡na_{n}=\dfrac {1}{(\ln \ln n)^{\ln n}} (n>1)(n>1).

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Problem 2618

Investigate the convergence of the series with the following general term: an=1(ln⁡n)ln⁡ln⁡na_{n}=\dfrac {1}{(\ln n)^{\ln \ln n}} (n>1)(n>1).

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Problem 2619

Using the Cauchy integral test, investigate the convergence of the series with the following general term: an=1nln⁡pna_{n}=\dfrac {1}{n\ln^{p} n}.

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Problem 2620

Using the Cauchy integral test, investigate the convergence of the series with the following general term: an=1n(ln⁡n)p(ln⁡ln⁡n)qa_{n}=\dfrac {1}{n(\ln n)^{p}(\ln \ln n)^{q}} (n>2)(n>2).

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Problem 2620.1
[NO TASK]
Problem 2620.1
[NO TASK]
Problem 2620.2
[NO TASK]
Problem 2620.2
[NO TASK]
Problem 2620.3
[NO TASK]
Problem 2620.3
[NO TASK]
Problem 2621

Investigate the convergence of the series

∑n=2∞1ln⁡(n!) \sum _{n=2}^{\infty } \frac{1}{\ln (n!)}
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Problem 2622

Prove that if the terms of the positive series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} decrease monotonically, then the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} and the series ∑n=1∞2na2n\displaystyle \sum_{n=1}^{\infty } 2^{n} a_{2^{n}} converge or diverge together.

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Problem 2623

Let f(x)f(x) be a monotonically nonincreasing positive-valued function. Prove that if the series ∑n=1∞f(n)\displaystyle \sum_{n=1}^{\infty } f(n) converges, then for its remainder Rn=∑k=n+1∞f(k)\displaystyle R_{n}=\sum_{k=n+1}^{\infty } f(k) we have the following estimate:

∫n+1+∞f(x) dx<Rn<f(n+1)+∫n+1+∞f(x) dx. \int _{n+1}^{+\infty } f(x)\, dx<R_{n}<f(n+1)+\int _{n+1}^{+\infty } f(x)\, dx .

Using this formula, find the sum of the series ∑n=1∞1n3\displaystyle \sum_{n=1}^{\infty } \frac{1}{n^{3}} to within 0.010.01.

?
Problem 2624

Prove Yermakov's test: let f(x)f(x) be a monotonically decreasing positive-valued function, and let

lim⁡x→+∞exf(ex)f(x)=λ. \lim _{x \rightarrow +\infty } \frac{e^{x} f(e^{x})}{f(x)}=\lambda .

If λ<1\lambda <1, then the series ∑n=1∞f(n)\displaystyle \sum_{n=1}^{\infty } f(n) converges; if λ>1\lambda >1, then the series ∑n=1∞f(n)\displaystyle \sum_{n=1}^{\infty } f(n) diverges.

?
Problem 2625

Prove the Lobachevsky test: let the terms of the positive series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} decrease monotonically to zero; then the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} and the series

∑m=0∞pm2−m \sum _{m=0}^{\infty } p_{m} 2^{-m}

converge or diverge together, where pmp_{m} is the largest index of the terms ana_{n} satisfying the inequality

an≥2−m(n=1,2,…,pm) a_{n} \geq 2^{-m} \quad (n=1,2,\ldots ,p_{m})
?
Problem 2626

Investigate the convergence of the following series: ∑n=2∞n+2−n−2nα\displaystyle \sum_{n=2}^{\infty } \frac{\sqrt{n+2}-\sqrt{n-2}}{n^{\alpha }}.

?
Problem 2627

Investigate the convergence of the following series: ∑n=1∞(n+a−n2+n+b4)\displaystyle \sum_{n=1}^{\infty }\left(\sqrt{n+a}-\sqrt[4]{n^{2}+n+b}\right).

?
Problem 2628

Investigate the convergence of the following series: ∑n=1∞(ctg⁡nπ4n−2−sin⁡nπ2n+1)\displaystyle \sum_{n=1}^{\infty }\left(\ctg \frac{n\pi }{4n-2}-\sin \frac{n\pi }{2n+1}\right).

?
Problem 2629

Investigate the convergence of the following series:∑n=1∞(1n−ln⁡n+1n)\displaystyle \sum_{n=1}^{\infty }\left(\frac{1}{\sqrt{n}}-\sqrt{\ln \frac{n+1}{n}}\right).

?
Problem 2630

Investigate the convergence of the following series:∑n=1∞ln⁡(n!)na\displaystyle \sum_{n=1}^{\infty } \frac{\ln (n!)}{n^{a}}.

?
Problem 2631

Investigate the convergence of the following series:∑n=1∞e−n3\displaystyle \sum_{n=1}^{\infty } e^{-\sqrt[3]{n}}.

?
Problem 2632

Investigate the convergence of the following series:∑n=1∞n2e−n\displaystyle \sum_{n=1}^{\infty } n^{2} e^{-\sqrt{n}}.

?
Problem 2633

Investigate the convergence of the following series:∑n=1∞(n1n2+1−1)\displaystyle \sum_{n=1}^{\infty }\left(n^{\frac{1}{n^{2}+1}}-1\right).

?
Problem 2634

Investigate the convergence of the following series:∑n=1∞ealn⁡n+bcln⁡n+d\displaystyle \sum_{n=1}^{\infty } e^{\frac{a\ln n+b}{c\ln n+d}}.

?
Problem 2635

Investigate the convergence of the following series:∑n=1∞1ln⁡2(sin⁡1n)\displaystyle \sum_{n=1}^{\infty } \frac{1}{\ln^{2}\left(\sin \dfrac {1}{n}\right)}.

?
Problem 2636

Investigate the convergence of the following series:∑n=1∞(cos⁡an)n3\displaystyle \sum_{n=1}^{\infty }\left(\cos \frac{a}{n}\right)^{n^{3}}.

?
Problem 2637

Investigate the convergence of the following series:∑n=1∞ln⁡(ch⁡πncos⁡πn)\displaystyle \sum_{n=1}^{\infty } \ln \left(\dfrac {\ch \dfrac {\pi }{n}}{\cos \dfrac {\pi }{n}}\right).

?
Problem 2638

Investigate the convergence of the following series:∑n=1∞n!nn\displaystyle \sum_{n=1}^{\infty } \frac{n!}{n^{\sqrt{n}}}.

?
Problem 2639

Investigate the convergence of the following series: ∑n=2∞nln⁡n(ln⁡n)n\displaystyle \sum_{n=2}^{\infty } \frac{n^{\ln n}}{(\ln n)^{n}}.

?
Problem 2640

Investigate the convergence of the following series: ∑n=1∞(a1n−b1n+c1n2)\displaystyle \sum_{n=1}^{\infty }\left(a^{\frac{1}{n}}-\frac{b^{\frac{1}{n}}+c^{\frac{1}{n}}}{2}\right) (a>0,b>0,c>0)(a>0, b>0, c>0).

?
Problem 2641

Investigate the convergence of the following series: ∑n=1∞(nna−1)\displaystyle \sum_{n=1}^{\infty }\left(n^{n^{a}}-1\right).

?
Problem 2642

Investigate the convergence of the following series:∑n=1∞[ln⁡1na−ln⁡(sin⁡1na)]\displaystyle \sum_{n=1}^{\infty }\left[\ln \frac{1}{n^{a}}-\ln \left(\sin \frac{1}{n^{a}}\right)\right].

?
Problem 2643

Investigate the convergence of the following series:∑n=1∞a−(bln⁡n+cln⁡2n)(a>0)\displaystyle \sum_{n=1}^{\infty } a^{-(b\ln n+c\ln^{2} n)} \quad (a>0).

?
Problem 2644

Investigate the convergence of the following series:∑n=1∞n2n(n+a)n+b(n+b)n+a(a>0,b>0)\displaystyle \sum_{n=1}^{\infty } \frac{n^{2n}}{(n+a)^{n+b}(n+b)^{n+a}} \quad (a>0, b>0).

?
Problem 2645

Investigate the convergence of the following series:∑n=1∞[(n+1)!]n2!⋅4!⋯(2n)!\displaystyle \sum_{n=1}^{\infty } \frac{\left[(n+1)!\right]^{n}}{2! \cdot 4! \cdots (2n)!}.

?
Problem 2646

Study the convergence of the series ∑n=1∞un\displaystyle \sum_{n=1}^{\infty } u_{n}, whose general term is: un=∫01nx dx1+x2u_{n}=\displaystyle \int_{0}^{\frac{1}{n}} \frac{\sqrt{x}\, dx}{1+x^{2}}.

?
Problem 2647

Study the convergence of the series ∑n=1∞un\displaystyle \sum_{n=1}^{\infty } u_{n}, whose general term is: un=1∫0n1+x44 dxu_{n}=\dfrac {1}{\displaystyle \int_{0}^{n} \sqrt[4]{1+x^{4}}\, dx}.

?
Problem 2648

Study the convergence of the series ∑n=1∞un\displaystyle \sum_{n=1}^{\infty } u_{n}, whose general term is: un=∫nπ(n+1)πsin⁡2xx dxu_{n}=\displaystyle \int_{n\pi }^{(n+1)\pi } \frac{\sin^{2} x}{x}\, dx.

?
Problem 2649

Study the convergence of the series ∑n=1∞un\displaystyle \sum_{n=1}^{\infty } u_{n}, whose general term is: un=∫nn+1e−x dxu_{n}=\displaystyle \int_{n}^{n+1} e^{-\sqrt{x}}\, dx.

?
Problem 2650

Study the convergence of the series ∑n=1∞un\displaystyle \sum_{n=1}^{\infty } u_{n}, whose general term is: un=∫0πnsin⁡3x1+x dxu_{n}=\displaystyle \int_{0}^{\frac{\pi }{n}} \frac{\sin^{3} x}{1+x}\, dx.

?
Problem 2651

Study the convergence of the series ∑n=1∞un\displaystyle \sum_{n=1}^{\infty } u_{n}, whose general term is: un=1!+2!+⋯+n!(2n)!u_{n}=\dfrac {1!+2!+\cdots +n!}{(2n)!}.

?
Problem 2652

Study the convergence of the series ∑n=1∞un\displaystyle \sum_{n=1}^{\infty } u_{n}, whose general term is: un=∑k=1nln⁡2knau_{n}=\dfrac {\displaystyle \sum_{k=1}^{n} \ln^{2} k}{n^{a}}.

?
Problem 2653

Replace the sequence xnx_{n} (n=1,2,…)(n=1,2,\ldots ) by the corresponding series, and then study its convergence, where: xn=1+12+⋯+1n−2nx_{n}=1+\dfrac {1}{\sqrt{2}}+\cdots +\dfrac {1}{\sqrt{n}}-2\sqrt{n}.

?
Problem 2654

Replace the sequence xnx_{n} (n=1,2,…)(n=1,2,\ldots ) by the corresponding series, and then study its convergence, where: xn=∑k=1nln⁡kk−(ln⁡n)22x_{n}=\displaystyle \sum_{k=1}^{n} \frac{\ln k}{k}-\frac{(\ln n)^{2}}{2}.

?
Problem 2655

If

?
(a)

∑n=1∞1n2\displaystyle \sum_{n=1}^{\infty } \frac{1}{n^{2}};

(b)

∑n=1∞2n(n+1)!\displaystyle \sum_{n=1}^{\infty } \frac{2^{n}}{(n+1)!} ∗)^{*)};

(c)

∑n=1∞1(2n−1)!\displaystyle \sum_{n=1}^{\infty } \frac{1}{(2n-1)!},

approximately how many terms of the series must be taken in order to compute the sum of the series accurate to within 10−510^{-5}.

Remark.
?

A problem number carrying a "+" superscript at the upper right indicates that the answer given here does not agree with the answer given in the appendix of the original Chinese translation of the problem collection; this will not be noted again below. The Chinese translation was based essentially on the second Russian edition, and some errors in the second Russian edition have since been corrected in the third Russian edition.

Section 5.2Tests for the convergence of series with variable sign[48/94%]
Problem 2656

Prove: the terms of a non-absolutely convergent series can, without changing their order, be grouped into blocks so that the resulting new series converges absolutely.

?
Problem 2657

Let there be a series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n}; if (a) as n→∞n \rightarrow \infty, the general term ana_{n} of this series tends to zero; (б) a certain series ∑n=1∞An\displaystyle \sum_{n=1}^{\infty } A_{n}, obtained by combining the terms of the given series without changing their original order, converges; (в) in the terms An=∑i=pnpn+1−1ai\displaystyle A_{n}=\sum_{i=p_{n}}^{p_{n+1}-1} a_{i} (p1<p2<⋯ )(p_{1}<p_{2}<\cdots ), the number of terms aia_{i} being summed is bounded, prove that the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges.

?
Problem 2658

Prove: if the terms of a convergent series are rearranged so that each term is displaced from its original position by no more than mm positions (mm being a number given in advance), then its sum is unchanged.

?
Problem 2659

Prove the convergence of the following series and find its sum: 1−32+54−78+⋯ .1-\dfrac {3}{2}+\dfrac {5}{4}-\dfrac {7}{8}+\cdots .

?
Problem 2660

Prove the convergence of the following series and find its sum: 1+12−14+18+116−132+⋯ .1+\dfrac {1}{2}-\dfrac {1}{4}+\dfrac {1}{8}+\dfrac {1}{16}-\dfrac {1}{32}+\cdots .

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Problem 2661

Prove the convergence of the following series and find its sum: 1−12+13−14+15−16+⋯ .1-\dfrac {1}{2}+\dfrac {1}{3}-\dfrac {1}{4}+\dfrac {1}{5}-\dfrac {1}{6}+\cdots .

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Problem 2662

Given ∑n=1∞(−1)n+1n=ln⁡2\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n}=\ln 2, find the sum of the series obtained by rearranging the terms of the given series in the order:

?
(a)

1+13−12+15+17−14+⋯1+\dfrac {1}{3}-\dfrac {1}{2}+\dfrac {1}{5}+\dfrac {1}{7}-\dfrac {1}{4}+\cdots

(b)

1−12−14+13−16−18+⋯1-\dfrac {1}{2}-\dfrac {1}{4}+\dfrac {1}{3}-\dfrac {1}{6}-\dfrac {1}{8}+\cdots

Problem 2663

Rearrange the terms of the convergent series

∑n=1∞(−1)n+1n \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{\sqrt{n}}

so that it becomes divergent.

?
Problem 2664

Investigate the convergence of the variable-sign series: ∑n=1∞(−1)n(n−1)22n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{\frac{n(n-1)}{2}}}{2^{n}}.

?
Problem 2665

Investigate the convergence of the variable-sign series: ∑n=1∞(−1)n(2n+1003n+1)n\displaystyle \sum_{n=1}^{\infty }(-1)^{n}\left(\frac{2n+100}{3n+1}\right)^{n}.

?
Problem 2666

Investigate the convergence of the variable-sign series: 1+12+13−14−15−16+17+18+19−⋯ .1+\dfrac {1}{2}+\dfrac {1}{3}-\dfrac {1}{4}-\dfrac {1}{5}-\dfrac {1}{6}+\dfrac {1}{7}+\dfrac {1}{8}+\dfrac {1}{9}-\cdots .

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Problem 2666.1
[NO TASK]
Problem 2666.1
[NO TASK]
Problem 2667

Investigate the convergence of the variable-sign series: ∑n=1∞ln⁡100nnsin⁡nπ4\displaystyle \sum_{n=1}^{\infty } \frac{\ln^{100} n}{n}\sin \frac{n\pi }{4}.

?
Problem 2668

Investigate the convergence of the variable-sign series: ∑n=1∞(−1)nsin⁡2nn\displaystyle \sum_{n=1}^{\infty }(-1)^{n} \frac{\sin^{2} n}{n}.

?
Problem 2669

Investigate the convergence of the variable-sign series: ∑n=1∞(−1)nnn+100\displaystyle \sum_{n=1}^{\infty }(-1)^{n} \frac{\sqrt{n}}{n+100}.

?
Problem 2670

Investigate the convergence of the variable-sign series: ∑n=2∞(−1)nn+(−1)n\displaystyle \sum_{n=2}^{\infty } \frac{(-1)^{n}}{\sqrt{n}+(-1)^{n}}.

?
Problem 2671

Investigate the convergence of the variable-sign series: ∑n=1∞sin⁡(πn2+k2)\displaystyle \sum_{n=1}^{\infty } \sin \left(\pi \sqrt{n^{2}+k^{2}}\right).

?
Problem 2672

Investigate the convergence of the variable-sign series: ∑n=1∞(−1)[n]n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{\left[\sqrt{n}\right]}}{n}.

?
Problem 2673

Investigate the convergence of the variable-sign series: ∑n=1∞(−1)nnn\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{\sqrt[n]{n}}.

?
Problem 2673.1
[NO TASK]
Problem 2673.1
[NO TASK]
Problem 2674

Prove: if

lim⁡n→∞n(bnbn+1−1)>0, \lim _{n \rightarrow \infty } n\left(\frac{b_{n}}{b_{n+1}}-1\right)>0,

then the alternating series b1−b2+b3−b4+⋯+(−1)n−1bn+⋯b_{1}-b_{2}+b_{3}-b_{4}+\cdots +(-1)^{n-1}b_{n}+\cdots (bn>0)(b_{n}>0) converges.

?
Problem 2675

Investigate the absolute and conditional convergence of the series: ∑n=1∞(−1)n−1np\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n^{p}}.

?
Problem 2676

Investigate the absolute and conditional convergence of the series: ∑n=1∞(−1)n−1np+1n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n^{p+\frac{1}{n}}}.

?
Problem 2677

Investigate the absolute and conditional convergence of the following series: ∑n=2∞ln⁡[1+(−1)nnp]\displaystyle \sum_{n=2}^{\infty } \ln \left[1+\frac{(-1)^{n}}{n^{p}}\right].

?
Problem 2678

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)n−12nsin⁡2nxn\displaystyle \sum_{n=1}^{\infty }(-1)^{n-1} \frac{2^{n} \sin^{2n} x}{n}.

?
Problem 2679

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)nx+n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{x+n}.

?
Problem 2680

Investigate the absolute and conditional convergence of the following series: ∑n=2∞(−1)n[n+(−1)n]p\displaystyle \sum_{n=2}^{\infty } \frac{(-1)^{n}}{\left[n+(-1)^{n}\right]^{p}}.

?
Problem 2681

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)n−1[n+(−1)n−1]p\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{\left[\sqrt{n}+(-1)^{n-1}\right]^{p}}.

?
Problem 2682

Investigate the absolute and conditional convergence of the following series: ∑n=1∞sin⁡nπ4np+sin⁡nπ4\displaystyle \sum_{n=1}^{\infty } \frac{\sin \dfrac {n\pi }{4}}{n^{p}+\sin \dfrac {n\pi }{4}}.

?
Problem 2683

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)nn−1n+11n100\displaystyle \sum_{n=1}^{\infty }(-1)^{n} \frac{n-1}{n+1} \frac{1}{\sqrt[100]{n}}.

?
Problem 2684

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)n(n−1)2⋅n1002n\displaystyle \sum_{n=1}^{\infty }(-1)^{\frac{n(n-1)}{2}} \cdot \frac{n^{100}}{2^{n}}.

?
Problem 2685

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)nn2n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{n^{2}\sqrt{n}}.

?
Problem 2686

Investigate the absolute and conditional convergence of the following series: ∑n=2∞sin⁡nπ12ln⁡n\displaystyle \sum_{n=2}^{\infty } \frac{\sin \dfrac {n\pi }{12}}{\ln n}.

?
Problem 2687

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)[n]np\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{\left[\sqrt{n}\right]}}{n^{p}}.

?
Problem 2688

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)[ln⁡n]n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{\left[\ln n\right]}}{n}.

?
Problem 2689

Investigate the absolute and conditional convergence of the following series: ∑n=1∞(−1)n−1(1⋅3⋅5⋯(2n−1)2⋅4⋅6⋯(2n))p\displaystyle \sum_{n=1}^{\infty }(-1)^{n-1}\left(\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}\right)^{p}.

?
Problem 2690

Investigate the absolute and conditional convergence of the following series: ∑n=1∞sin⁡n⋅sin⁡n2n\displaystyle \sum_{n=1}^{\infty } \frac{\sin n \cdot \sin n^{2}}{n}.

?
Problem 2691

Investigate the absolute and conditional convergence of the following series: ∑n=1∞sin⁡n2\displaystyle \sum_{n=1}^{\infty } \sin n^{2}.

?
Problem 2692

Let

R(x)=a0xp+a1xp−1+⋯+apb0xq+b1xq−1+⋯+bq R(x)=\frac{a_{0}x^{p}+a_{1}x^{p-1}+\cdots +a_{p}}{b_{0}x^{q}+b_{1}x^{q-1}+\cdots +b_{q}}

be a rational function, where a0≠0a_{0} \neq 0, b0≠0b_{0} \neq 0, and when x≥n0x \geq n_{0}, ∣b0xq+b1xq−1+⋯+bq∣>0\left|b_{0}x^{q}+b_{1}x^{q-1}+\cdots +b_{q}\right|>0.

Investigate the absolute and conditional convergence of the series

∑n=n0∞(−1)nR(n) \sum _{n=n_{0}}^{\infty }(-1)^{n} R(n)
?
Problem 2693

Investigate the convergence of the following series: 11p−12q+13p−14q+15p−16q+⋯ .\dfrac {1}{1^{p}}-\dfrac {1}{2^{q}}+\dfrac {1}{3^{p}}-\dfrac {1}{4^{q}}+\dfrac {1}{5^{p}}-\dfrac {1}{6^{q}}+\cdots .

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Problem 2694

Investigate the convergence of the following series: 1+13p−12p+15p+17p−14p+⋯ .1+\dfrac {1}{3^{p}}-\dfrac {1}{2^{p}}+\dfrac {1}{5^{p}}+\dfrac {1}{7^{p}}-\dfrac {1}{4^{p}}+\cdots .

?
Problem 2695

Investigate the convergence of the series: 1+13p−11p+15p+17p−13p+19p+111p−15p+⋯ .1+\dfrac {1}{3^{p}}-\dfrac {1}{1^{p}}+\dfrac {1}{5^{p}}+\dfrac {1}{7^{p}}-\dfrac {1}{3^{p}}+\dfrac {1}{9^{p}}+\dfrac {1}{11^{p}}-\dfrac {1}{5^{p}}+\cdots .

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Problem 2696

Investigate the convergence of the series: 1−22q+13p+14p−25q+16p+17p−28q+19p+⋯ .1-\dfrac {2}{2^{q}}+\dfrac {1}{3^{p}}+\dfrac {1}{4^{p}}-\dfrac {2}{5^{q}}+\dfrac {1}{6^{p}}+\dfrac {1}{7^{p}}-\dfrac {2}{8^{q}}+\dfrac {1}{9^{p}}+\cdots .

?
Problem 2697

Prove: the series

?
(a)

sin⁡x+sin⁡2x2+sin⁡3x3+⋯\sin x+\dfrac {\sin 2x}{2}+\dfrac {\sin 3x}{3}+\cdots,

(b)

cos⁡x+cos⁡2x2+cos⁡3x3+⋯\cos x+\dfrac {\cos 2x}{2}+\dfrac {\cos 3x}{3}+\cdots

are not absolutely convergent on the interval (0,π)(0,\pi ).

Problem 2698

For the series

∑n=1∞cos⁡nxnp,∑n=1∞sin⁡nxnp(0<x<π) \sum _{n=1}^{\infty } \frac{\cos nx}{n^{p}}, \quad \sum _{n=1}^{\infty } \frac{\sin nx}{n^{p}} \quad (0<x<\pi )

determine, for all parameters (p,x)(p,x): (a) the region of absolute convergence; (б) the region of non-absolute convergence.

?
(a)
(b)
Problem 2698.1
(a)
(b)
(c)
Problem 2698.1
(a)
(b)
(c)
Problem 2699

For the series

∑n=1∞(−1)n−1(1+p)(2+p)⋯(n+p)n!nq \sum _{n=1}^{\infty }(-1)^{n-1} \frac{(1+p)(2+p)\cdots (n+p)}{n!n^{q}}

determine:

?
(a)

the region of absolute convergence;

(b)

the region of conditional convergence.

Problem 2700

Investigate the convergence of the series

∑n=1∞(mn) \sum _{n=1}^{\infty } \binom {m}{n}

where (mn)=m(m−1)⋯(m−n+1)n!\dbinom {m}{n}=\dfrac {m(m-1)\cdots (m-n+1)}{n!}.

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Section 5.3Operations on Series[16/94%]
Problem 2701

If the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges and lim⁡n→∞bnan=1\displaystyle \lim_{n \rightarrow \infty } \frac{b_{n}}{a_{n}}=1, can we conclude that the series ∑n=1∞bn\displaystyle \sum_{n=1}^{\infty } b_{n} also converges?

?
Problem 2702

Let ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} be a non-absolutely convergent series, and

Pn=∑i=1n∣ai∣+ai2,Nn=∑i=1n∣ai∣−ai2, P_{n}=\sum _{i=1}^{n} \frac{\left|a_{i}\right|+a_{i}}{2}, \quad N_{n}=\sum _{i=1}^{n} \frac{\left|a_{i}\right|-a_{i}}{2},

prove that lim⁡n→∞NnPn=1\displaystyle \lim_{n \rightarrow \infty } \frac{N_{n}}{P_{n}}=1.

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Problem 2703

Prove: for every p>0p>0, the sum of the series

∑n=1∞(−1)n+1np \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{n^{p}}

lies between 12\dfrac {1}{2} and 11.

?
Problem 2703.1
(a)
(b)
Problem 2703.1
(a)
(b)
Problem 2704

Prove: if the terms of the series

1−12+13−14+15−⋯ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots

are rearranged so that groups of pp consecutive positive terms alternate with groups of qq consecutive negative terms, then the sum of the new series is

ln⁡2+12ln⁡pq. \ln 2+\frac{1}{2}\ln \frac{p}{q}.
?
Problem 2705

Prove: if the signs of the terms of the harmonic series 1+12+13+14+⋯1+\dfrac {1}{2}+\dfrac {1}{3}+\dfrac {1}{4}+\cdots are changed so that pp positive terms are followed by qq negative terms (p≠qp \neq q), without otherwise changing the original order, then this series is always divergent. It is convergent only when p=qp=q.

?
Problem 2706

If, of two series: (a) one series converges while the other diverges; (б) both series diverge, what can be said about the sum of the two series?

?
(a)
(b)
Problem 2707

Find the sum of the two series: ∑n=1∞(13n+(−1)nn3)+∑n=1∞(13n+1+(−1)n+1n3)\displaystyle \sum_{n=1}^{\infty } \left(\frac{1}{3^{n}}+\frac{(-1)^{n}}{n^{3}}\right)+\sum_{n=1}^{\infty } \left(\frac{1}{3^{n+1}}+\frac{(-1)^{n+1}}{n^{3}}\right).

?
Problem 2708

Find the sum of the series: ∑n=1∞(12n+(−1)n3n)\displaystyle \sum_{n=1}^{\infty } \left(\frac{1}{2^{n}}+\frac{(-1)^{n}}{3^{n}}\right).

?
Problem 2709

Find the sum of the series: ∑n=1∞cos⁡2nπ32n\displaystyle \sum_{n=1}^{\infty } \frac{\cos \dfrac {2n\pi }{3}}{2^{n}}.

?
Problem 2710

Find the sum of the series: ∑n=0∞x⌊n/2⌋y⌊(n+1)/2⌋(∣xy∣<1)\displaystyle \sum_{n=0}^{\infty } x^{\left\lfloor n/2\right\rfloor } y^{\left\lfloor (n+1)/2\right\rfloor } \quad (\left|xy\right|<1).

?
Problem 2711

Prove that ∑n=0∞1n!⋅∑n=0∞(−1)nn!=1\displaystyle \sum_{n=0}^{\infty } \frac{1}{n!} \cdot \sum_{n=0}^{\infty } \frac{(-1)^{n}}{n!}=1.

?
Problem 2712

Prove that (∑n=0∞qn)2=∑n=0∞(n+1)qn(∣q∣<1)\displaystyle \left(\sum_{n=0}^{\infty } q^{n}\right)^{2}=\sum_{n=0}^{\infty } (n+1)q^{n} \quad (\left|q\right|<1).

?
Problem 2713

Prove that the square of the convergent series ∑n=1∞(−1)n+1n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{\sqrt{n}} is a divergent series.

?
Problem 2714

Prove that the product of the two series ∑n=1∞(−1)n−1nα (α>0)\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n^{\alpha }} \ (\alpha >0) and ∑n=1∞(−1)n−1nβ (β>0)\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n^{\beta }} \ (\beta >0) is a convergent series when α+β>1\alpha +\beta >1, and a divergent series when α+β<1\alpha +\beta <1.

?
Problem 2715

Verify that the product of the two divergent series 1−∑n=1∞(32)n1-\displaystyle \sum_{n=1}^{\infty } \left(\frac{3}{2}\right)^{n} and 1+∑n=1∞(32)n−1(2n+12n+1)1+\displaystyle \sum_{n=1}^{\infty } \left(\frac{3}{2}\right)^{n-1}\left(2^{n}+\frac{1}{2^{n+1}}\right) is an absolutely convergent series.

?
Section 5.4Function series[99/97%]
Problem 2716

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞nxn\displaystyle \sum_{n=1}^{\infty } \frac{n}{x^{n}}.

?
Problem 2717

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞(−1)n2n−1(1−x1+x)n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{2n-1}\left(\frac{1-x}{1+x}\right)^{n}.

?
Problem 2718

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞nn+1(x2x+1)n\displaystyle \sum_{n=1}^{\infty } \frac{n}{n+1}\left(\frac{x}{2x+1}\right)^{n}.

?
Problem 2719

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞1⋅3⋯(2n−1)2⋅4⋯(2n)(2x1+x2)n\displaystyle \sum_{n=1}^{\infty } \frac{1 \cdot 3 \cdots (2n-1)}{2 \cdot 4 \cdots (2n)}\left(\frac{2x}{1+x^{2}}\right)^{n}.

?
Problem 2720

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞n⋅32n2nxn(1−x)n\displaystyle \sum_{n=1}^{\infty } \frac{n \cdot 3^{2n}}{2^{n}} x^{n}(1-x)^{n}.

?
Problem 2721

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞2nsin⁡nxn2\displaystyle \sum_{n=1}^{\infty } \frac{2^{n}\sin^{n} x}{n^{2}}.

?
Problem 2722

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞(−1)n(x+n)p\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{(x+n)^{p}}.

?
Problem 2723

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞npsin⁡nx1+nq (q>0;0<x<π)\displaystyle \sum_{n=1}^{\infty } \frac{n^{p}\sin nx}{1+n^{q}} \ (q>0; 0<x<\pi ).

?
Problem 2724

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞xn1−xn\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{1-x^{n}} (Lambert series).

?
Problem 2725

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞[x(x+n)n]n\displaystyle \sum_{n=1}^{\infty } \left[\frac{x(x+n)}{n}\right]^{n}.

?
Problem 2726

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞xn1+x2n\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{1+x^{2n}}.

?
Problem 2727

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞xn(1+x)(1+x2)⋯(1+xn)\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{(1+x)(1+x^{2})\cdots (1+x^{n})}.

?
Problem 2728

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞ne−nx\displaystyle \sum_{n=1}^{\infty } n e^{-nx}.

?
Problem 2729

Determine the domain of absolute and conditional convergence of the following function series: ∑n=1∞1n!n⋅11+a2nx2\displaystyle \sum_{n=1}^{\infty } \frac{1}{\sqrt[n]{n!}} \cdot \frac{1}{1+a^{2n}x^{2}}.

?
Problem 2730

Determine the domain of absolute and conditional convergence of the following functional series: ∑n=1∞(2−x)(2−x12)(2−x13)⋯(2−x1n) (x>0)\displaystyle \sum_{n=1}^{\infty } (2-x)\left(2-x^{\frac{1}{2}}\right)\left(2-x^{\frac{1}{3}}\right)\cdots \left(2-x^{\frac{1}{n}}\right) \ (x>0).

?
Problem 2731

Determine the domain of absolute and conditional convergence of the following functional series: ∑n=1∞(n+x)nnn+x\displaystyle \sum_{n=1}^{\infty } \frac{(n+x)^{n}}{n^{n+x}}.

?
Problem 2732

Determine the domain of absolute and conditional convergence of the following functional series: ∑n=1∞xnynxn+yn (x>0,y>0)\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}y^{n}}{x^{n}+y^{n}} \ (x>0, y>0).

?
Problem 2733

Determine the domain of absolute and conditional convergence of the following functional series: ∑n=1∞xnn+yn (y≥0)\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{n+y^{n}} \ (y \geq 0).

?
Problem 2734

Determine the domain of absolute and conditional convergence of the following functional series: ∑n=1∞∣x∣n2+∣y∣n2n\displaystyle \sum_{n=1}^{\infty } \sqrt[n]{\left|x\right|^{n^{2}}+\left|y\right|^{n^{2}}}.

?
Problem 2735

Determine the domain of absolute and conditional convergence of the following functional series: ∑n=1∞ln⁡(1+xn)ny (x≥0)\displaystyle \sum_{n=1}^{\infty } \frac{\ln (1+x^{n})}{n^{y}} \ (x \geq 0).

?
Problem 2736

Determine the domain of absolute and conditional convergence of the following functional series: ∑n=1∞tan⁡n(x+yn)\displaystyle \sum_{n=1}^{\infty } \tan^{n}\left(x+\frac{y}{n}\right).

?
Problem 2737

Prove: if the Laurent series ∑n=−∞+∞anxn\displaystyle \sum_{n=-\infty }^{+\infty } a_{n}x^{n} converges at x=x1x=x_{1} and x=x2 (∣x1∣<∣x2∣)x=x_{2} \ (\left|x_{1}\right|<\left|x_{2}\right|), then this series also converges when ∣x1∣<∣x∣<∣x2∣\left|x_{1}\right|<\left|x\right|<\left|x_{2}\right|.

?
Problem 2738

Find the domain of convergence of the Laurent series ∑n=−∞+∞n2∣n∣xn\displaystyle \sum_{n=-\infty }^{+\infty } \frac{n}{2^{\left|n\right|}} x^{n} and find its sum.

?
Problem 2739

Determine the domain of absolute and conditional convergence of the Newton series:

?
(a)

∑n=1∞x[n]n!\displaystyle \sum_{n=1}^{\infty } \frac{x^{[n]}}{n!};

(b)

∑n=1∞1np⋅x[n]n!\displaystyle \sum_{n=1}^{\infty } \frac{1}{n^{p}} \cdot \frac{x^{[n]}}{n!};

(c)

∑n=1∞(ex)ny[n]nn\displaystyle \sum_{n=1}^{\infty } \frac{(ex)^{n} y^{[n]}}{n^{n}}.

where x[n]=x(x−1)⋯[x−(n−1)]x^{[n]}=x(x-1)\cdots \left[x-(n-1)\right].

Problem 2740

Prove: if the Dirichlet series ∑n=1∞annx\displaystyle \sum_{n=1}^{\infty } \frac{a_{n}}{n^{x}} converges at x=x0x=x_{0}, then this series also converges when x>x0x>x_{0}.

?
Problem 2741

Prove: the necessary and sufficient condition for the sequence fn(x) (n=1,2,…)f_{n}(x) \ (n=1,2,\ldots ) to converge uniformly on the interval (a,b)(a,b) to the limit function f(x)f(x) is

lim⁡n→∞{sup⁡a<x<bγn(x)}=0, \lim _{n \rightarrow \infty } \left\{ \sup _{a<x<b} \gamma _{n}(x)\right\} =0,

where γn(x)=∣f(x)−fn(x)∣\gamma_{n}(x)=\left|f(x)-f_{n}(x)\right|.

?
Problem 2742

What does it mean for the sequence fn(x) (n=1,2,…)f_{n}(x) \ (n=1,2,\ldots ): (a) to converge on the interval (x0,+∞)(x_{0},+\infty ); (b) to converge uniformly on every finite interval (a,b)⊂(x0,+∞)(a,b) \subset (x_{0},+\infty ); (c) to converge uniformly on the interval (x0,+∞)(x_{0},+\infty )?

?
(a)
(b)
(c)
Problem 2743

For the sequence fn(x)=xn (n=1,2,…) (0<x<1)f_{n}(x)=x^{n} \ (n=1,2,\ldots ) \ (0<x<1), find the smallest index N=N(ε,x)N=N(\varepsilon ,x) of its terms, such that from this term on the terms of the sequence differ from the limit function at the given point xx by no more than 0.0010.001, where x=110,110,…,110m,…x=\dfrac {1}{10}, \dfrac {1}{\sqrt{10}}, \ldots , \dfrac {1}{\sqrt[m]{10}}, \ldots.

Does this sequence converge uniformly on the given interval (0,1)(0,1)?

?
Problem 2744

How many terms of the series ∑n=1∞sin⁡nxn(n+1)\displaystyle \sum_{n=1}^{\infty } \frac{\sin nx}{n(n+1)} should be taken so that the partial sum Sn(x)S_{n}(x) differs from the sum of the series by less than ε\varepsilon, for −∞<x<+∞-\infty <x<+\infty? Given:

?
(a)

ε=0.1\varepsilon =0.1;

(b)

ε=0.01\varepsilon =0.01;

(c)

ε=0.001\varepsilon =0.001.

Find the value of nn.

Problem 2745

For which nn can the inequality

∣ex−∑i=0nxii!∣<0.001(0≤x≤10) \left|e^{x}-\sum _{i=0}^{n} \frac{x^{i}}{i!}\right|<0.001 \quad (0 \leq x \leq 10)

be guaranteed to hold?

?
Problem 2746

Investigate the uniform convergence of the sequence fn(x)=xnf_{n}(x)=x^{n} on the given interval:

?
(a)

0≤x≤120 \leq x \leq \dfrac {1}{2};

(b)

0≤x≤10 \leq x \leq 1.

Problem 2747

Investigate the uniform convergence of the sequence fn(x)=xn−xn+1f_{n}(x)=x^{n}-x^{n+1} on the indicated interval 0≤x≤10 \leq x \leq 1.

?
Problem 2748

Investigate the uniform convergence of the sequence fn(x)=xn−x2nf_{n}(x)=x^{n}-x^{2n} on the indicated interval 0≤x≤10 \leq x \leq 1.

?
Problem 2749

Investigate the uniform convergence of the sequence fn(x)=1x+nf_{n}(x)=\dfrac {1}{x+n} on the indicated interval 0<x<+∞0<x<+\infty.

?
Problem 2750

Investigate the uniform convergence of the sequence fn(x)=nx1+n+xf_{n}(x)=\dfrac {nx}{1+n+x} on the indicated interval 0≤x≤10 \leq x \leq 1.

?
Problem 2751

Investigate the uniform convergence of the sequence fn(x)=xn1+xnf_{n}(x)=\dfrac {x^{n}}{1+x^{n}} on the indicated intervals:

?
(a)

0≤x≤1−ε0 \leq x \leq 1-\varepsilon;

(b)

1−ε≤x≤1+ε1-\varepsilon \leq x \leq 1+\varepsilon;

(c)

1+ε≤x<+∞1+\varepsilon \leq x<+\infty,

where ε>0\varepsilon >0.

Problem 2752

Investigate the uniform convergence of the sequence fn(x)=2nx1+n2x2f_{n}(x)=\dfrac {2nx}{1+n^{2}x^{2}} on the indicated intervals:

?
(a)

0≤x≤10 \leq x \leq 1;

(b)

1<x<+∞1<x<+\infty.

Problem 2753

Investigate the uniform convergence of the sequence fn(x)=x2+1n2f_{n}(x)=\sqrt{x^{2}+\dfrac {1}{n^{2}}} on the indicated interval −∞<x<+∞-\infty <x<+\infty.

?
Problem 2754

Investigate the uniform convergence of the sequence fn(x)=n(x+1n−x)f_{n}(x)=n\left(\sqrt{x+\dfrac {1}{n}}-\sqrt{x}\right) on the indicated interval 0<x<+∞0<x<+\infty.

?
Problem 2755

Investigate the uniform convergence of the sequences on the indicated interval −∞<x<+∞-\infty <x<+\infty:

?
(a)

fn(x)=sin⁡nxnf_{n}(x)=\dfrac {\sin nx}{n};

(b)

fn(x)=sin⁡xnf_{n}(x)=\sin \dfrac {x}{n}.

Problem 2756

Investigate the uniform convergence of the sequences on the indicated interval 0<x<+∞0<x<+\infty:

?
(a)

fn(x)=arctan⁡nxf_{n}(x)=\arctan nx;

(b)

fn(x)=xarctan⁡nxf_{n}(x)=x\arctan nx.

Problem 2757

Investigate the uniform convergence of the sequence fn(x)=en(x−1)f_{n}(x)=e^{n(x-1)} on the indicated interval 0<x<10<x<1.

?
Problem 2758

Investigate the uniform convergence of the sequence fn(x)=e−(x−n)2f_{n}(x)=e^{-(x-n)^{2}} on the indicated intervals:

?
(a)

−l<x<l-l<x<l, where ll is an arbitrary positive number;

(b)

−∞<x<+∞-\infty <x<+\infty.

Problem 2759

Investigate the uniform convergence of the sequence fn(x)=xnln⁡xnf_{n}(x)=\dfrac {x}{n}\ln \dfrac {x}{n} on the indicated interval 0<x<10<x<1.

?
Problem 2760

Investigate the uniform convergence of the sequence fn(x)=(1+xn)nf_{n}(x)=\left(1+\dfrac {x}{n}\right)^{n} on the indicated intervals:

?
(a)

on a finite interval (a,b)(a,b);

(b)

on the interval (−∞,+∞)(-\infty ,+\infty ).

Problem 2761

Investigate the uniform convergence of the sequence fn(x)=n(x1n−1)f_{n}(x)=n\left(x^{\frac{1}{n}}-1\right) on the indicated interval 1≤x≤a1 \leq x \leq a.

?
Problem 2762

Investigate the uniform convergence of the sequence fn(x)=1+xnnf_{n}(x)=\sqrt[n]{1+x^{n}} on the indicated interval 0≤x≤20 \leq x \leq 2.

?
Problem 2763

Investigate the uniform convergence of the sequence

fn(x)={n2x,if 0≤x≤1n;n2(2n−x),if 1n<x<2n;0,if x≥2n f_{n}(x)= \begin{cases} n^{2}x, & \text{if } 0 \leq x \leq \dfrac {1}{n}; \\ n^{2}\left(\dfrac {2}{n}-x\right), & \text{if } \dfrac {1}{n}<x<\dfrac {2}{n}; \\ 0, & \text{if } x \geq \dfrac {2}{n} \end{cases}

on the interval 0≤x≤10 \leq x \leq 1.

?
Problem 2764

Let f(x)f(x) be an arbitrary function defined on the interval (a,b)(a,b), and let fn(x)=[nf(x)]n (n=1,2,…)f_{n}(x)=\dfrac {\left[nf(x)\right]}{n} \ (n=1,2,\ldots ). Prove that as n→∞n \rightarrow \infty, fn(x)\unifconvf(x) (a<x<b)f_{n}(x) \unifconv f(x) \ (a<x<b).

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Problem 2765

Let the function f(x)f(x) have a continuous derivative f′(x)f'(x) on the interval (a,b)(a,b), and let

fn(x)=n[f(x+1n)−f(x)]. f_{n}(x)=n\left[f\left(x+\frac{1}{n}\right)-f(x)\right].

Prove that on the closed interval α≤x≤β\alpha \leq x \leq \beta (where a<α<β<ba<\alpha <\beta <b), fn(x)\unifconvf′(x)f_{n}(x) \unifconv f'(x).

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Problem 2766

Let fn(x)=∑i=0n−11nf(x+in)f_{n}(x)=\displaystyle \sum_{i=0}^{n-1} \frac{1}{n}f\left(x+\frac{i}{n}\right), where f(x)f(x) is a continuous function. Prove that the sequence fn(x)f_{n}(x) converges uniformly on every finite closed interval [a,b]\left[a,b\right].

?
Problem 2767

Investigate the convergence of the following series: ∑n=0∞xn\displaystyle \sum_{n=0}^{\infty } x^{n}:

?
(a)

on the interval ∣x∣<q\left|x\right|<q, where q<1q<1;

(b)

on the interval ∣x∣<1\left|x\right|<1.

Problem 2768

Investigate the convergence of the following series: ∑n=1∞xnn2\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{n^{2}}, on the closed interval −1≤x≤1-1 \leq x \leq 1.

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Problem 2768.1
[NO TASK]
Problem 2768.1
[NO TASK]
Problem 2769

Investigate the convergence of the following series: ∑n=0∞(1−x)xn\displaystyle \sum_{n=0}^{\infty } (1-x)x^{n}, on the closed interval 0≤x≤10 \leq x \leq 1.

?
Problem 2770

Investigate the convergence of the following series: ∑n=1∞(xnn−xn+1n+1)\displaystyle \sum_{n=1}^{\infty } \left(\frac{x^{n}}{n}-\frac{x^{n+1}}{n+1}\right), on the closed interval −1≤x≤1-1 \leq x \leq 1.

?
Problem 2771

Investigate the convergence of the following series: ∑n=1∞x[(n−1)x+1](nx+1)\displaystyle \sum_{n=1}^{\infty } \frac{x}{\left[(n-1)x+1\right](nx+1)}, 0<x<+∞0<x<+\infty.

?
Problem 2772

Investigate the convergence of the following series: ∑n=1∞1(x+n)(x+n+1)\displaystyle \sum_{n=1}^{\infty } \frac{1}{(x+n)(x+n+1)}, 0<x<+∞0<x<+\infty.

?
Problem 2773

Investigate the convergence of the following series: ∑n=1∞nx(1+x)(1+2x)⋯(1+nx)\displaystyle \sum_{n=1}^{\infty } \frac{nx}{(1+x)(1+2x)\cdots (1+nx)}:

?
(a)

0≤x≤ε0 \leq x \leq \varepsilon, where ε>0\varepsilon >0;

(b)

ε≤x≤+∞\varepsilon \leq x \leq +\infty.

Problem 2774

Using the Weierstrass test, prove the uniform convergence of the following functional series on the indicated intervals:

?
(a)

∑n=1∞1x2+n2\displaystyle \sum_{n=1}^{\infty } \frac{1}{x^{2}+n^{2}}, −∞<x<+∞-\infty <x<+\infty;

(b)

∑n=1∞(−1)nx+2n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{x+2^{n}}, −2<x<+∞-2<x<+\infty;

(c)

∑n=1∞x1+n4x2\displaystyle \sum_{n=1}^{\infty } \frac{x}{1+n^{4}x^{2}}, 0≤x≤+∞0 \leq x \leq +\infty;

(d)

∑n=1∞nx1+n5x2\displaystyle \sum_{n=1}^{\infty } \frac{nx}{1+n^{5}x^{2}}, ∣x∣<+∞\left|x\right|<+\infty;

(e)

∑n=1∞n2n!(xn+x−n)\displaystyle \sum_{n=1}^{\infty } \frac{n^{2}}{\sqrt{n!}}\left(x^{n}+x^{-n}\right), 12≤∣x∣≤2\dfrac {1}{2} \leq \left|x\right| \leq 2;

(f)

∑n=1∞xn(n2)!\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{\left(\dfrac {n}{2}\right)!}, ∣x∣<a\left|x\right|<a, aa an arbitrary positive number;

(g)

∑n=1∞sin⁡nxn4+x43\displaystyle \sum_{n=1}^{\infty } \frac{\sin nx}{\sqrt[3]{n^{4}+x^{4}}}, ∣x∣<+∞\left|x\right|<+\infty;

(h)

∑n=1∞cos⁡nxn2\displaystyle \sum_{n=1}^{\infty } \frac{\cos nx}{n^{2}}, ∣x∣<+∞\left|x\right|<+\infty;

(i)

∑n=1∞sin⁡nxnn\displaystyle \sum_{n=1}^{\infty } \frac{\sin nx}{n\sqrt{n}}, ∣x∣<+∞\left|x\right|<+\infty;

(j)

∑n=2∞ln⁡(1+xnln⁡2n)\displaystyle \sum_{n=2}^{\infty } \ln \left(1+\frac{x}{n\ln^{2} n}\right), ∣x∣<a\left|x\right|<a;

(k)

∑n=1∞x2e−nx\displaystyle \sum_{n=1}^{\infty } x^{2}e^{-nx}, 0≤x<+∞0 \leq x<+\infty;

(l)

∑n=1∞arctan⁡2xx2+n3\displaystyle \sum_{n=1}^{\infty } \arctan \frac{2x}{x^{2}+n^{3}}, ∣x∣<+∞\left|x\right|<+\infty.

Problem 2775

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=1∞sin⁡nxn\displaystyle \sum_{n=1}^{\infty } \frac{\sin nx}{n}:

?
(a)

on the closed interval ε≤x≤2π−ε\varepsilon \leq x \leq 2\pi -\varepsilon, where ε>0\varepsilon >0;

(b)

on the closed interval 0≤x≤2π0 \leq x \leq 2\pi.

Problem 2776

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=1∞2nsin⁡13nx\displaystyle \sum_{n=1}^{\infty } 2^{n}\sin \dfrac {1}{3^{n}x}, 0<x<+∞0<x<+\infty.

?
Problem 2777

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=1∞(−1)nx+n\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{x+n}, 0<x<+∞0<x<+\infty.

?
Problem 2778

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=2∞(−1)nn+sin⁡x\displaystyle \sum_{n=2}^{\infty } \frac{(-1)^{n}}{n+\sin x}, 0≤x≤2π0 \leq x \leq 2\pi.

?
Problem 2779

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=1∞(−1)n(n−1)2n2+ex3\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{\frac{n(n-1)}{2}}}{\sqrt[3]{n^{2}+e^{x}}}, ∣x∣≤10\left|x\right| \leq 10.

?
Problem 2780

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=1∞cos⁡2nπ3n2+x2\displaystyle \sum_{n=1}^{\infty } \frac{\cos \dfrac {2n\pi }{3}}{\sqrt{n^{2}+x^{2}}}, −∞<x<+∞-\infty <x<+\infty.

?
Problem 2781

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=1∞sin⁡xsin⁡nxn+x\displaystyle \sum_{n=1}^{\infty } \frac{\sin x \sin nx}{\sqrt{n+x}}, 0≤x<+∞0 \leq x<+\infty.

?
Problem 2782

Investigate the uniform convergence of the following functional series on the indicated interval: ∑n=1∞(−1)[n]n(n+x)\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{\left[\sqrt{n}\right]}}{\sqrt{n(n+x)}}, 0≤x<+∞0 \leq x<+\infty.

?
Problem 2783

Can a sequence of discontinuous functions converge uniformly to a continuous function?

?
Problem 2784

Prove: if the series ∑n=1∞∣fn(x)∣\displaystyle \sum_{n=1}^{\infty } \left|f_{n}(x)\right| converges uniformly on [a,b]\left[a,b\right], then the series ∑n=1∞fn(x)\displaystyle \sum_{n=1}^{\infty } f_{n}(x) also converges uniformly on [a,b]\left[a,b\right].

?
Problem 2785

If the series ∑n=1∞fn(x)\displaystyle \sum_{n=1}^{\infty } f_{n}(x) converges absolutely and uniformly on [a,b]\left[a,b\right], must the series ∑n=1∞∣fn(x)∣\displaystyle \sum_{n=1}^{\infty } \left|f_{n}(x)\right| necessarily converge uniformly on [a,b]\left[a,b\right]?

Investigate the example ∑n=1∞(−1)n(1−x)xn\displaystyle \sum_{n=1}^{\infty } (-1)^{n}(1-x)x^{n}, where 0≤x≤10 \leq x \leq 1.

?
Problem 2786

Prove that the series

∑n=1∞fn(x)(0≤x≤1), \sum _{n=1}^{\infty } f_{n}(x) \quad (0 \leq x \leq 1),

where

fn(x)={0,if 0≤x≤2−(n+1);1nsin⁡2(2n+1πx),if 2−(n+1)<x<2−n;0,if 2−n≤x≤1, f_{n}(x)= \begin{cases} 0, & \text{if } 0 \leq x \leq 2^{-(n+1)}; \\ \dfrac {1}{n}\sin ^{2}(2^{n+1}\pi x), & \text{if } 2^{-(n+1)}<x<2^{-n}; \\ 0, & \text{if } 2^{-n} \leq x \leq 1, \end{cases}

converges absolutely and uniformly, admits no convergent series of positive constant terms as a dominating (majorant) series.

?
Problem 2787

Prove that if a series ∑n=1∞φn(x)\displaystyle \sum_{n=1}^{\infty } \varphi_{n}(x), each of whose terms is a monotone function, converges absolutely at the endpoints of the closed interval [a,b]\left[a,b\right], then this series converges absolutely and uniformly on the closed interval [a,b]\left[a,b\right].

?
Problem 2788

Prove that the power series ∑n=0∞anxn\displaystyle \sum_{n=0}^{\infty } a_{n}x^{n} converges uniformly on any closed interval lying entirely within its interval of convergence.

?
Problem 2789

Suppose an→∞a_{n} \rightarrow \infty and the series ∑n=1∞∣1an∣\displaystyle \sum_{n=1}^{\infty } \left|\dfrac {1}{a_{n}}\right| converges. Prove that the series ∑n=1∞1x−an\displaystyle \sum_{n=1}^{\infty } \dfrac {1}{x-a_{n}} converges absolutely and uniformly on any bounded closed set not containing any of the points an (n=1,2,…)a_{n} \ (n=1,2,\ldots ).

?
Problem 2790

Prove that if the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges, then the Dirichlet series ∑n=1∞annx\displaystyle \sum_{n=1}^{\infty } \dfrac {a_{n}}{n^{x}} converges uniformly for x≥0x \geq 0.

?
Problem 2791

Suppose the series ∑n=1∞an\displaystyle \sum_{n=1}^{\infty } a_{n} converges. Prove that the series ∑n=1∞ane−nx\displaystyle \sum_{n=1}^{\infty } a_{n}e^{-nx} converges uniformly on the domain x≥0x \geq 0.

?
Problem 2792

Prove that the function f(x)=∑n=1∞sin⁡nxn3f(x)=\displaystyle \sum_{n=1}^{\infty } \dfrac {\sin nx}{n^{3}} is continuous and has a continuous derivative on the domain −∞<x<+∞-\infty <x<+\infty.

?
Problem 2793

Prove that the function f(x)=∑n=−∞+∞1(n−x)2f(x)=\displaystyle \sum_{n=-\infty }^{+\infty } \dfrac {1}{(n-x)^{2}}:

?
(a)

is defined and continuous at every point except x=0,±1,±2,…x=0, \pm 1, \pm 2, \ldots;

(b)

is a periodic function with period 11.

Problem 2794

Prove that the series ∑n=1∞[nxe−nx−(n−1)xe−(n−1)x]\displaystyle \sum_{n=1}^{\infty } \left[nxe^{-nx}-(n-1)xe^{-(n-1)x}\right] converges on the closed interval 0≤x≤10 \leq x \leq 1 but not uniformly, while its sum is a continuous function on this interval.

?
Problem 2795

Determine the domain of existence of the function f(x)f(x) and investigate its continuity, given:

?
(a)

f(x)=∑n=1∞(x+1n)nf(x)=\displaystyle \sum_{n=1}^{\infty } \left(x+\dfrac {1}{n}\right)^{n};

(b)

f(x)=∑n=1∞x+(−1)nnx2+n2f(x)=\displaystyle \sum_{n=1}^{\infty } \dfrac {x+(-1)^{n}n}{x^{2}+n^{2}};

(c)

f(x)=∑n=1∞x(1+x2)nf(x)=\displaystyle \sum_{n=1}^{\infty } \dfrac {x}{(1+x^{2})^{n}}.

Problem 2796

Let rkr_{k} (k=1,2,…)(k=1,2,\ldots ) be the rational numbers of the interval [0,1]\left[0,1\right]. Prove that the function

f(x)=∑k=1∞∣x−rk∣3k(0≤x≤1) f(x)=\sum _{k=1}^{\infty } \frac{\left|x-r_{k}\right|}{3^{k}} \quad (0 \leq x \leq 1)

has the following properties:

?
(a)

it is continuous;

(b)

it is differentiable at the irrational points and not differentiable at the rational points.

Problem 2797

Prove that the Riemann ζ\zeta function ζ(x)=∑n=1∞1nx\zeta (x)=\displaystyle \sum_{n=1}^{\infty } \dfrac {1}{n^{x}} is continuous on the domain x>1x>1 and has continuous derivatives of all orders on this domain.

?
Problem 2798

Prove that the θ\theta function θ(x)=∑n=−∞+∞e−n2x\theta (x)=\displaystyle \sum_{n=-\infty }^{+\infty } e^{-n^{2}x} is defined for x>0x>0 and is infinitely differentiable there.

?
Problem 2799

Determine the domain of existence of the function f(x)f(x) and investigate its differentiability, given:

?
(a)

f(x)=∑n=1∞(−1)nxn+xf(x)=\displaystyle \sum_{n=1}^{\infty } \dfrac {(-1)^{n}x}{n+x};

(b)

f(x)=∑n=1∞∣x∣n2+x2f(x)=\displaystyle \sum_{n=1}^{\infty } \dfrac {\left|x\right|}{n^{2}+x^{2}}.

Problem 2800

Prove that the sequence fn(x)=1narctan⁡xn (n=1,2,…)f_{n}(x)=\dfrac {1}{n}\arctan x^{n} \ (n=1,2,\ldots ) converges uniformly on the interval (−∞,+∞)(-\infty ,+\infty ), but

[lim⁡n→∞fn(x)]x=1′≠lim⁡n→∞fn′(1). \left[\lim _{n \rightarrow \infty } f_{n}(x)\right]'_{x=1} \neq \lim _{n \rightarrow \infty } f_{n}'(1).
?
Problem 2801

Prove that the sequence fn(x)=x2+1nsin⁡n(x+π2)f_{n}(x)=x^{2}+\dfrac {1}{n}\sin n\left(x+\dfrac {\pi }{2}\right) converges uniformly on the interval (−∞,+∞)(-\infty ,+\infty ), but

[lim⁡n→∞fn(x)]′≠lim⁡n→∞fn′(x). \left[\lim _{n \rightarrow \infty } f_{n}(x)\right]' \neq \lim _{n \rightarrow \infty } f_{n}'(x).
?
Problem 2802

For what values of the parameter α\alpha: (a) does the sequence

fn(x)=nαxe−nx(n=1,2,…) f_{n}(x)=n^{\alpha }xe^{-nx} \quad (n=1,2,\ldots )

converge on the closed interval [0,1]\left[0,1\right]; (б) does the sequence in (a) converge uniformly on [0,1]\left[0,1\right]; (в) can lim⁡n→∞∫01fn(x)dx\displaystyle \lim_{n \rightarrow \infty } \int_{0}^{1} f_{n}(x)dx be taken by passing the limit under the integral sign?

?
(a)
(b)
(c)
Problem 2803

Prove that the sequence fn(x)=nxe−nx2 (n=1,2,…)f_{n}(x)=nxe^{-nx^{2}} \ (n=1,2,\ldots ) converges on the interval [0,1]\left[0,1\right], but

∫01[lim⁡n→∞fn(x)]dx≠lim⁡n→∞∫01fn(x)dx. \int _{0}^{1} \left[\lim _{n \rightarrow \infty } f_{n}(x)\right]dx \neq \lim _{n \rightarrow \infty } \int _{0}^{1} f_{n}(x)dx.
?
Problem 2804

Prove that the sequence fn(x)=nx(1−x)n (n=1,2,…)f_{n}(x)=nx(1-x)^{n} \ (n=1,2,\ldots ) converges but does not converge uniformly on the closed interval [0,1]\left[0,1\right], yet

lim⁡n→∞∫01fn(x)dx=∫01[lim⁡n→∞fn(x)]dx. \lim _{n \rightarrow \infty } \int _{0}^{1} f_{n}(x)dx=\int _{0}^{1} \left[\lim _{n \rightarrow \infty } f_{n}(x)\right]dx.
?
Problem 2805

Is it valid to pass the limit under the integral sign in the following expression

lim⁡n→∞∫01nx1+n2x4dx \lim _{n \rightarrow \infty } \int _{0}^{1} \frac{nx}{1+n^{2}x^{4}}dx

?

?
Problem 2806

Compute: lim⁡x→1−0∑n=1∞(−1)n+1n⋅xnxn+1\displaystyle \lim_{x \rightarrow 1-0} \sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n} \cdot \frac{x^{n}}{x^{n}+1}.

?
Problem 2807

Compute: lim⁡x→1−0∑n=1∞(xn−xn+1)\displaystyle \lim_{x \rightarrow 1-0} \sum_{n=1}^{\infty } (x^{n}-x^{n+1}).

?
Problem 2808

Compute: lim⁡x→+0∑n=1∞12nnx\displaystyle \lim_{x \rightarrow +0} \sum_{n=1}^{\infty } \frac{1}{2^{n}n^{x}}.

?
Problem 2808.1
[NO TASK]
Problem 2808.1
[NO TASK]
Problem 2809

Is it valid to differentiate the series ∑n=1∞arctan⁡xn2\displaystyle \sum_{n=1}^{\infty } \arctan \dfrac {x}{n^{2}} term by term?

?
Problem 2810

Is it valid to integrate the series

∑n=1∞(x12n+1−x12n−1) \sum _{n=1}^{\infty } \left(x^{\frac{1}{2n+1}}-x^{\frac{1}{2n-1}}\right)

term by term on the closed interval [0,1]\left[0,1\right]?

?
Problem 2811

Let f(x) (−∞<x<+∞)f(x) \ (-\infty <x<+\infty ) be a function differentiable any number of times, and suppose the sequence of its derivatives f(n)(x) (n=1,2,…)f^{(n)}(x) \ (n=1,2,\ldots ) converges uniformly on every finite interval (a,b)(a,b) to a function φ(x)\varphi (x). Prove that φ(x)=Cex\varphi (x)=Ce^{x}, where CC is a constant.

?
Problem 2811.1
[NO TASK]
Problem 2811.1
[NO TASK]
Section 5.5Power Series[130/96%]
Problem 2812

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞xnnp\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{n^{p}}.

?
Problem 2813

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞3n+(−2)nn(x+1)n\displaystyle \sum_{n=1}^{\infty } \frac{3^{n}+(-2)^{n}}{n}(x+1)^{n}.

?
Problem 2814

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(n!)2(2n)!xn\displaystyle \sum_{n=1}^{\infty } \frac{(n!)^{2}}{(2n)!}x^{n}.

?
Problem 2815

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞αn2⋅xn (0<α<1)\displaystyle \sum_{n=1}^{\infty } \alpha^{n^{2}} \cdot x^{n} \ (0<\alpha <1).

?
Problem 2816

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(1+1n)n2xn\displaystyle \sum_{n=1}^{\infty } \left(1+\frac{1}{n}\right)^{n^{2}}x^{n}.

?
Problem 2817

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞n!an2xn (a>1)\displaystyle \sum_{n=1}^{\infty } \frac{n!}{a^{n^{2}}}x^{n} \ (a>1).

?
Problem 2818

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(1⋅3⋅5⋯(2n−1)2⋅4⋅6⋯(2n))p(x−12)n\displaystyle \sum_{n=1}^{\infty } \left(\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}\right)^{p}\left(\frac{x-1}{2}\right)^{n}.

?
Problem 2819

Find the radius and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(−1)n[2n(n!)2(2n+1)!]pxn\displaystyle \sum_{n=1}^{\infty } (-1)^{n}\left[\frac{2^{n}(n!)^{2}}{(2n+1)!}\right]^{p}x^{n}.

?
Problem 2820

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞m(m−1)⋯(m−n+1)n!xn\displaystyle \sum_{n=1}^{\infty } \frac{m(m-1) \cdots (m-n+1)}{n!}x^{n}.

?
Problem 2821

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(ann+bnn2)xn (a>0,b>0)\displaystyle \sum_{n=1}^{\infty } \left(\frac{a^{n}}{n}+\frac{b^{n}}{n^{2}}\right)x^{n} \ (a>0,b>0).

?
Problem 2822

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞xnan+bn (a>0,b>0)\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{a^{n}+b^{n}} \ (a>0,b>0).

?
Problem 2823

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞xnan (a>0)\displaystyle \sum_{n=1}^{\infty } \frac{x^{n}}{a^{\sqrt{n}}} \ (a>0).

?
Problem 2824

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞3−nxnn2+1\displaystyle \sum_{n=1}^{\infty } \frac{3^{-\sqrt{n}}x^{n}}{\sqrt{n^{2}+1}}.

?
Problem 2825

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(2n)!!(2n+1)!!xn\displaystyle \sum_{n=1}^{\infty } \frac{(2n)!!}{(2n+1)!!}x^{n}.

?
Problem 2826

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(−1)nn!(ne)nxn\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n}}{n!}\left(\frac{n}{e}\right)^{n}x^{n}.

?
Problem 2827

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(1+12+⋯+1n)xn\displaystyle \sum_{n=1}^{\infty } \left(1+\frac{1}{2}+\cdots +\frac{1}{n}\right)x^{n}.

?
Problem 2828

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞[3+(−1)n]nnxn\displaystyle \sum_{n=1}^{\infty } \frac{\left[3+(-1)^{n}\right]^{n}}{n}x^{n}.

?
Problem 2829

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=2∞(1+2cos⁡πn4)nln⁡nxn\displaystyle \sum_{n=2}^{\infty } \frac{\left(1+2\cos \dfrac {\pi n}{4}\right)^{n}}{\ln n}x^{n}.

?
Problem 2830

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞xn22n\displaystyle \sum_{n=1}^{\infty } \frac{x^{n^{2}}}{2^{n}}.

?
Problem 2831

Find the radius of convergence and interval of convergence of the following power series, and study its behavior at the endpoints of the interval of convergence: ∑n=1∞(−1)[n]nxn\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{\left[\sqrt{n}\right]}}{n}x^{n} (the Pringsheim series).

?
Problem 2831.1
[NO TASK]
Problem 2831.1
[NO TASK]
Problem 2831.2
[NO TASK]
Problem 2831.2
[NO TASK]
Problem 2832

Find the domain of convergence of the hypergeometric series

1+α⋅β1⋅γx+α(α+1)β(β+1)1⋅2⋅γ(γ+1)x2+⋯+α(α+1)⋯(α+n−1)β(β+1)⋯(β+n−1)1⋅2⋯n⋅γ(γ+1)⋯(γ+n−1)xn+⋯ 1+\frac{\alpha \cdot \beta }{1 \cdot \gamma }x+\frac{\alpha (\alpha +1)\beta (\beta +1)}{1 \cdot 2 \cdot \gamma (\gamma +1)}x^{2}+\cdots +\frac{\alpha (\alpha +1) \cdots (\alpha +n-1)\beta (\beta +1) \cdots (\beta +n-1)}{1 \cdot 2 \cdots n \cdot \gamma (\gamma +1) \cdots (\gamma +n-1)}x^{n}+\cdots
?
Problem 2833

Find the domain of convergence of the following generalized power series: ∑n=0∞12n+1(1−x1+x)n\displaystyle \sum_{n=0}^{\infty } \frac{1}{2n+1}\left(\frac{1-x}{1+x}\right)^{n}.

?
Problem 2834

Find the domain of convergence of the following generalized power series: ∑n=1∞1xnsin⁡π2n\displaystyle \sum_{n=1}^{\infty } \frac{1}{x^{n}}\sin \frac{\pi }{2^{n}}.

?
Problem 2835

Find the domain of convergence of the following generalized power series: ∑n=−∞+∞xn2n2\displaystyle \sum_{n=-\infty }^{+\infty } \frac{x^{n}}{2^{n^{2}}}.

?
Problem 2836

Find the domain of convergence of the following generalized power series: ∑n=1∞(1+1n)−n2e−nx\displaystyle \sum_{n=1}^{\infty } \left(1+\frac{1}{n}\right)^{-n^{2}}e^{-nx}.

?
Problem 2837

Find the domain of convergence of the following generalized power series: ∑n=1∞33n(n!)3(3n)!tan⁡nx\displaystyle \sum_{n=1}^{\infty } \frac{3^{3n}(n!)^{3}}{(3n)!}\tan^{n}x.

?
Problem 2838

Expand the function f(x)=x3f(x)=x^{3} in nonnegative integer powers of the binomial x+1x+1.

?
Problem 2839

Expand the function f(x)=1a−x (a≠0)f(x)=\dfrac {1}{a-x} \ (a \neq 0) into a power series in each of the following ways: (a)(a) in powers of xx; (б)(б) in powers of the binomial x−bx-b, where b≠ab \neq a; (в)(в) in powers of 1x\dfrac {1}{x}. Find the corresponding domains of convergence.

?
(a)
(b)
(c)
Problem 2840

Expand the function f(x)=ln⁡xf(x)=\ln x in positive integer powers of the difference x−1x-1, and state the interval of convergence of the expansion. Find the sum of the series ∑n=1∞(−1)n+1n\displaystyle \sum_{n=1}^{\infty } \dfrac {(-1)^{n+1}}{n}.

?
Problem 2841

Write out the expansion of the function f(x)=sh⁡xf(x)=\operatorname {sh}x in positive integer powers of the variable xx, and find the corresponding interval of convergence.

?
Problem 2842

Write out the expansion of the function f(x)=ch⁡xf(x)=\operatorname {ch}x in positive integer powers of the variable xx, and find the corresponding interval of convergence.

?
Problem 2843

Write out the expansion of the function f(x)=sin⁡2xf(x)=\sin^{2}x in positive integer powers of the variable xx, and find the corresponding interval of convergence.

?
Problem 2844

Write out the expansion of the function f(x)=ax (a>0)f(x)=a^{x} \ (a>0) in positive integer powers of the variable xx, and find the corresponding interval of convergence.

?
Problem 2845

Write the expansion of the function f(x)=sin⁡(uarcsin⁡x)f(x)=\sin (u\arcsin x) in nonnegative integer powers of the variable xx, and find the corresponding interval of convergence.

?
Problem 2846

Write the expansion of the function f(x)=cos⁡(uarcsin⁡x)f(x)=\cos (u\arcsin x) in nonnegative integer powers of the variable xx, and find the corresponding interval of convergence.

?
Problem 2847

Write the first three terms of the expansion of the function f(x)=xxf(x)=x^{x} in nonnegative integer powers of the difference x−1x-1.

?
Problem 2848

Write the first three terms of the expansion of the function f(x)=(1+x)1x (x≠0)f(x)=(1+x)^{\frac{1}{x}} \ (x \neq 0) and f(0)=ef(0)=e in nonnegative integer powers of the variable xx.

?
Problem 2849

Expand the functions sin⁡(x+h)\sin (x+h) and cos⁡(x+h)\cos (x+h) in nonnegative integer powers of the variable hh.

?
Problem 2850

Without actually carrying out the expansion, find the interval of convergence of the power series expansion of the function f(x)=xx2−5x+6f(x)=\dfrac {x}{x^{2}-5x+6}: (a)(a) expanded in powers of xx; (б)(б) expanded in powers of the binomial x−5x-5.

?
(a)
(b)
Problem 2850.1
[NO TASK]
Problem 2850.1
[NO TASK]
Problem 2851

Using the basic expansions I-V, write out the power series expansion in xx of the function e−x2e^{-x^{2}}.

?
Problem 2852

Using the basic expansions I-V, write out the power series expansion in xx of the function cos⁡2x\cos^{2}x.

?
Problem 2853

Using the basic expansions I-V, write out the power series expansion in xx of the function sin⁡3x\sin^{3}x.

?
Problem 2854

Using the basic expansions I-V, write out the power series expansion in xx of the function x101−x\dfrac {x^{10}}{1-x}.

?
Problem 2855

Using the basic expansions I-V, write out the power series expansion in xx of the function 1(1−x)2\dfrac {1}{(1-x)^{2}}.

?
Problem 2856

Using the basic expansions I-V, write out the power series expansion in xx of the function x1−2x\dfrac {x}{\sqrt{1-2x}}.

?
Problem 2857

Using the basic expansions I-V, write out the power series expansion in xx of the function ln⁡1+x1−x\ln \sqrt{\dfrac {1+x}{1-x}}.

?
Problem 2858

Using the basic expansions I-V, write out the power series expansion in xx of the function x1+x−2x2\dfrac {x}{1+x-2x^{2}}. (Hint: reduce the given fraction to its simplest rational fractions.)

?
Problem 2859

Using the basic expansions I-V, write out the power series expansion in xx of the function 12−5x6−5x−x2\dfrac {12-5x}{6-5x-x^{2}}.

?
Problem 2860

Using the basic expansions I-V, write out the power series expansion in xx of the function x(1−x)(1−x2)\dfrac {x}{(1-x)(1-x^{2})}.

?
Problem 2861

Using the basic expansions I-V, write out the power series expansion in xx of the function 11−x−x2\dfrac {1}{1-x-x^{2}}.

?
Problem 2862

Using the basic expansions I-V, write out the power series expansion in xx of the function 11+x+x2\dfrac {1}{1+x+x^{2}}.

?
Problem 2862.1

Let f(x)=11+x+x2+x3f(x)=\dfrac {1}{1+x+x^{2}+x^{3}}, find f1000(0)f^{1000}(0).

?
Problem 2862.1

Let f(x)=11+x+x2+x3f(x)=\dfrac {1}{1+x+x^{2}+x^{3}}, find f1000(0)f^{1000}(0).

?
Problem 2863

Using the basic expansions I-V, write out the power series expansion in xx of the function xcos⁡α−x21−2xcos⁡α+x2\dfrac {x\cos \alpha -x^{2}}{1-2x\cos \alpha +x^{2}}.

?
Problem 2864

Using the basic expansions I-V, write out the power series expansion in xx of the function xsin⁡α1−2xcos⁡α+x2\dfrac {x\sin \alpha }{1-2x\cos \alpha +x^{2}}.

?
Problem 2865

Using the basic expansions I-V, write out the power series expansion in xx of the function xsh⁡α1−2xch⁡α+x2\dfrac {x\operatorname {sh}\alpha }{1-2x\operatorname {ch}\alpha +x^{2}}.

?
Problem 2866

Using the basic expansions I-V, write out the power series expansion in xx of the function 1(1−x2)1−x2\dfrac {1}{(1-x^{2})\sqrt{1-x^{2}}}.

?
Problem 2867

Using the basic expansions I-V, write out the power series expansion in xx of the function ln⁡(1+x+x2+x3)\ln (1+x+x^{2}+x^{3}).

?
Problem 2868

Using the basic expansions I-V, write out the power series expansion in xx of the function excos⁡αcos⁡(xsin⁡α)e^{x\cos \alpha }\cos (x\sin \alpha ).

?
Problem 2869

First expand the derivative function, then use term-by-term integration to find the power series expansion of the function f(x)=arctan⁡xf(x)=\arctan x, and find the sum of the series ∑n=1∞(−1)n+12n−1\displaystyle \sum_{n=1}^{\infty } \dfrac {(-1)^{n+1}}{2n-1}.

?
Problem 2870

First expand the derivative function, then use term-by-term integration to find the power series expansion of the function f(x)=arcsin⁡xf(x)=\arcsin x.

?
Problem 2871

First expand the derivative function, then use term-by-term integration to find the power series expansion of the function f(x)=ln⁡(x+1+x2)f(x)=\ln (x+\sqrt{1+x^{2}}).

?
Problem 2872

First expand the derivative function, then use term-by-term integration to find the power series expansion of the function f(x)=ln⁡(1−2xcos⁡α+x2)f(x)=\ln (1-2x\cos \alpha +x^{2}).

?
Problem 2873

Using various methods, find the power series expansions of the following functions:

?
(a)

f(x)=(1+x)ln⁡(1+x)f(x)=(1+x)\ln (1+x);

(b)

f(x)=14ln⁡1+x1−x+12arctan⁡xf(x)=\dfrac {1}{4}\ln \dfrac {1+x}{1-x}+\dfrac {1}{2}\arctan x;

(c)

f(x)=arctan⁡2−2x1+4xf(x)=\arctan \dfrac {2-2x}{1+4x};

(d)

f(x)=arctan⁡2x2−x2f(x)=\arctan \dfrac {2x}{2-x^{2}};

(e)

f(x)=xarctan⁡x−ln⁡1+x2f(x)=x\arctan x-\ln \sqrt{1+x^{2}};

(f)

f(x)=arccos⁡(1−2x2)f(x)=\arccos (1-2x^{2});

(g)

f(x)=xarcsin⁡x+1−x2f(x)=x\arcsin x+\sqrt{1-x^{2}};

(h)

f(x)=xln⁡(x+1+x2)−1+x2f(x)=x\ln (x+\sqrt{1+x^{2}})-\sqrt{1+x^{2}}.

Problem 2874

Using the uniqueness of the expansion

f(x+h)−f(x)=hf′(x)+h22!f′′(x)+⋯ f(x+h)-f(x)=hf'(x)+\frac{h^{2}}{2!}f''(x)+\cdots

find the nn-th derivative of each of the following functions:

?
(a)

f(x)=ex2f(x)=e^{x^{2}};

(b)

f(x)=eaxf(x)=e^{\frac{a}{x}};

(c)

f(x)=arctan⁡xf(x)=\arctan x.

Problem 2875

Expand the function f(x)=ln⁡12+2x+x2f(x)=\ln \dfrac {1}{2+2x+x^{2}} in nonnegative integer powers of the binomial x+1x+1.

?
Problem 2876

Expand the function f(x)=11−xf(x)=\dfrac {1}{1-x} as a power series in negative integer powers of the variable xx.

?
Problem 2877

Expand the function f(x)=ln⁡xf(x)=\ln x as a power series in nonnegative integer powers of the fraction x−1x+1\dfrac {x-1}{x+1}.

?
Problem 2878

Expand the function f(x)=x1+xf(x)=\dfrac {x}{\sqrt{1+x}} as a power series in nonnegative integer powers of the fraction x1+x\dfrac {x}{1+x}.

?
Problem 2879

Let f(x)=∑n=0∞xnn!f(x)=\displaystyle \sum_{n=0}^{\infty } \dfrac {x^{n}}{n!}. Prove directly that f(x)f(y)=f(x+y)f(x)f(y)=f(x+y).

?
Problem 2880

Suppose we define sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!\sin x=\displaystyle \sum_{n=0}^{\infty } (-1)^{n}\dfrac {x^{2n+1}}{(2n+1)!} and cos⁡x=∑n=0∞(−1)nx2n(2n)!\cos x=\displaystyle \sum_{n=0}^{\infty } (-1)^{n}\dfrac {x^{2n}}{(2n)!}. Prove:

?
(a)

sin⁡xcos⁡x=12sin⁡2x\sin x\cos x=\dfrac {1}{2}\sin 2x;

(b)

sin⁡2x+cos⁡2x=1\sin^{2}x+\cos^{2}x=1.

Problem 2881

Write out several terms of the power series expansion of the function f(x)=(∑n=0∞(xnn+1))−1f(x)=\left(\displaystyle \sum_{n=0}^{\infty } \left(\dfrac {x^{n}}{n+1}\right)\right)^{-1}.

?
Problem 2882

Performing the corresponding operations on power series, expand the function f(x)=(1+x)e−xf(x)=(1+x)e^{-x} into a power series.

?
Problem 2883

Performing the corresponding operations on power series, expand the function f(x)=(1−x)2ch⁡xf(x)=(1-x)^{2}\operatorname {ch}\sqrt{x} into a power series.

?
Problem 2884

Performing the corresponding operations on power series, expand the function f(x)=ln⁡2(1−x)f(x)=\ln^{2}(1-x) into a power series.

?
Problem 2885

Performing the corresponding operations on power series, expand the function f(x)=(1+x2)arctan⁡xf(x)=(1+x^{2})\arctan x into a power series.

?
Problem 2886

Performing the corresponding operations on power series, expand the function f(x)=excos⁡xf(x)=e^{x}\cos x into a power series.

?
Problem 2887

Performing the corresponding operations on power series, expand the function f(x)=exsin⁡xf(x)=e^{x}\sin x into a power series.

?
Problem 2888

Performing the corresponding operations on power series, expand the function f(x)=ln⁡(1+x)1+xf(x)=\dfrac {\ln (1+x)}{1+x} into a power series.

?
Problem 2889

Performing the corresponding operations on power series, expand the function f(x)=(arctan⁡x)2f(x)=(\arctan x)^{2} into a power series.

?
Problem 2890

Performing the corresponding operations on power series, expand the function f(x)=(arcsin⁡xx)2f(x)=\left(\dfrac {\arcsin x}{x}\right)^{2} into a power series.

?
Problem 2891

Based on positive integer powers of the variable xx, write out the first three nonzero terms of the power series expansion of the function f(x)=tan⁡xf(x)=\tan x.

?
Problem 2892

Based on positive integer powers of the variable xx, write out the first three (nonzero) terms of the power series expansion of the function f(x)=th⁡xf(x)=\operatorname {th}x.

?
Problem 2893

Based on positive integer powers of the variable xx, write out the first three (nonzero) terms of the power series expansion of the function f(x)=cot⁡x−1xf(x)=\cot x-\dfrac {1}{x}.

?
Problem 2894

Let the expansion of sec⁡x\sec x be sec⁡x=∑n=0∞En(2n)!x2n\sec x=\displaystyle \sum_{n=0}^{\infty } \dfrac {E_{n}}{(2n)!}x^{2n}; derive a recurrence relation for the coefficients EnE_{n} (Euler numbers).

?
Problem 2895

Expand the function f(x)=11−2tx+x2f(x)=\dfrac {1}{\sqrt{1-2tx+x^{2}}} (∣x∣<1\left|x\right|<1) into a power series.

?
Problem 2896

Let f(x)=∑n=0∞anxnf(x)=\displaystyle \sum_{n=0}^{\infty } a_{n}x^{n}; write out the expansion of the function F(x)=f(x)1−xF(x)=\dfrac {f(x)}{1-x}.

?
Problem 2897

If the series ∑n=0∞anxn\displaystyle \sum_{n=0}^{\infty } a_{n}x^{n} has radius of convergence R1R_{1}, and the series ∑n=0∞bnxn\displaystyle \sum_{n=0}^{\infty } b_{n}x^{n} has radius of convergence R2R_{2}, what is the radius of convergence RR of the series

?
(a)

∑n=0∞(an+bn)xn\displaystyle \sum_{n=0}^{\infty } (a_{n}+b_{n})x^{n};

(b)

∑n=0∞anbnxn\displaystyle \sum_{n=0}^{\infty } a_{n}b_{n}x^{n}

?

Problem 2898

Let l=lim‾⁡n→∞∣anan+1∣l=\displaystyle \varliminf_{n \rightarrow \infty } \left|\dfrac {a_{n}}{a_{n+1}}\right|, L=lim‾⁡n→∞∣anan+1∣L=\displaystyle \varlimsup_{n \rightarrow \infty } \left|\dfrac {a_{n}}{a_{n+1}}\right|. Prove that the radius of convergence RR of the power series ∑n=0∞anxn\displaystyle \sum_{n=0}^{\infty } a_{n}x^{n} satisfies the inequality l≤R≤Ll \leq R \leq L.

?
Problem 2899

Prove that if the function f(x)=∑n=0∞an(x−x0)nf(x)=\displaystyle \sum_{n=0}^{\infty } a_{n}(x-x_{0})^{n} and ∣n!an∣<M\left|n!a_{n}\right|<M (n=1,2,…n=1,2,\ldots), where MM is a constant, then: 1) f(x)f(x) is infinitely differentiable at any point aa; 2) the following expansion holds

f(x)=∑n=0∞f(n)(a)n!(x−a)n(∣x∣<+∞). f(x)=\sum _{n=0}^{\infty } \frac{f^{(n)}(a)}{n!}(x-a)^{n} \quad (\left|x\right|<+\infty ).
?
Problem 2899.1
[NO TASK]
Problem 2899.1
[NO TASK]
Problem 2899.2
[NO TASK]
Problem 2899.2
[NO TASK]
Problem 2900

Prove that if 1) an≥0a_{n} \geq 0 and 2) lim⁡x→R−0∑n=0∞anxn=S\displaystyle \lim_{x \rightarrow R-0} \sum_{n=0}^{\infty } a_{n}x^{n}=S exists, then ∑n=0∞anRn=S\displaystyle \sum_{n=0}^{\infty } a_{n}R^{n}=S.

?
Problem 2901

Expand into a power series: ∫0xe−t2dt\displaystyle \int_{0}^{x} e^{-t^{2}}dt.

?
Problem 2902

Expand into a power series: ∫0xdt1−t4\displaystyle \int_{0}^{x} \dfrac {dt}{\sqrt{1-t^{4}}}.

?
Problem 2903

Expand into a power series: ∫0xsin⁡ttdt\displaystyle \int_{0}^{x} \dfrac {\sin t}{t}dt.

?
Problem 2904

Expand into a power series: ∫0xarctan⁡xxdx\displaystyle \int_{0}^{x} \dfrac {\arctan x}{x}dx.

?
Problem 2905

Expand into a power series, writing out four terms: ∫0xt dtln⁡(1+t)\displaystyle \int_{0}^{x} \dfrac {t\, dt}{\ln (1+t)}.

?
Problem 2906

Use term-by-term differentiation to compute the sum of the series x+x33+x55+⋯x+\dfrac {x^{3}}{3}+\dfrac {x^{5}}{5}+\cdots.

?
Problem 2907

Use term-by-term differentiation to compute the sum of the series x−x33+x55−⋯x-\dfrac {x^{3}}{3}+\dfrac {x^{5}}{5}-\cdots.

?
Problem 2908

Use term-by-term differentiation to compute the sum of the series 1+x22!+x44!+⋯1+\dfrac {x^{2}}{2!}+\dfrac {x^{4}}{4!}+\cdots.

?
Problem 2909

Use term-by-term differentiation to compute the sum of the series x1⋅2+x22⋅3+x33⋅4+⋯\dfrac {x}{1 \cdot 2}+\dfrac {x^{2}}{2 \cdot 3}+\dfrac {x^{3}}{3 \cdot 4}+\cdots.

?
Problem 2910

Use term-by-term differentiation to compute the sum of the series 1+12x+1⋅32⋅4x2+1⋅3⋅52⋅4⋅6x3+⋯1+\dfrac {1}{2}x+\dfrac {1 \cdot 3}{2 \cdot 4}x^{2}+\dfrac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}x^{3}+\cdots.

?
Problem 2911

Use term-by-term integration to compute the sum of the series x+2x2+3x3+⋯x+2x^{2}+3x^{3}+\cdots.

?
Problem 2912

Use term-by-term integration to compute the sum of the series x−4x2+9x3−16x4+⋯x-4x^{2}+9x^{3}-16x^{4}+\cdots.

?
Problem 2913

Use term-by-term integration to compute the sum of the series 1⋅2x+2⋅3x2+3⋅4x3+⋯1 \cdot 2x+2 \cdot 3x^{2}+3 \cdot 4x^{3}+\cdots.

?
Problem 2914

Prove that the series y=∑n=0∞x4n(4n)!y=\displaystyle \sum_{n=0}^{\infty } \dfrac {x^{4n}}{(4n)!} satisfies the equation y(4)=yy^{(4)}=y.

?
Problem 2915

Prove that the series y=∑n=0∞xn(n!)2y=\displaystyle \sum_{n=0}^{\infty } \dfrac {x^{n}}{(n!)^{2}} satisfies the equation xy′′+y′−y=0xy''+y'-y=0.

?
Problem 2916

Find, in the complex domain (z=x+iyz=x+iy), the radius and disk of convergence of the series ∑n=1∞(z−1−i)nn⋅2n\displaystyle \sum_{n=1}^{\infty } \dfrac {(z-1-i)^{n}}{n \cdot 2^{n}}.

?
Problem 2917

Find, in the complex domain (z=x+iyz=x+iy), the radius and disk of convergence of the series ∑n=1∞(1+i)nzn(n+1)(n+2)\displaystyle \sum_{n=1}^{\infty } \dfrac {(1+i)^{n}z^{n}}{(n+1)(n+2)}.

?
Problem 2918

Find, in the complex domain (z=x+iyz=x+iy), the radius and disk of convergence of the series ∑n=1∞n! zn(1+i)(1+2i)⋯(1+ni)\displaystyle \sum_{n=1}^{\infty } \dfrac {n!\, z^{n}}{(1+i)(1+2i)\cdots (1+ni)}.

?
Problem 2919

Find, in the complex domain (z=x+iyz=x+iy), the radius and disk of convergence of the series ∑n=1∞znnα+iβ\displaystyle \sum_{n=1}^{\infty } \dfrac {z^{n}}{n^{\alpha +i\beta }}.

?
Problem 2920

Find, in the complex domain (z=x+iyz=x+iy), the radius and disk of convergence of the series ∑n=1∞(z−eiα)nn(1−eiα)n\displaystyle \sum_{n=1}^{\infty } \dfrac {(z-e^{i\alpha })^{n}}{n(1-e^{i\alpha })^{n}}.

?
Problem 2921

Using Newton's binomial formula, approximately compute 93\sqrt[3]{9}, and estimate the error when only the first three terms of the expansion are taken.

?
Problem 2922

Compute approximately, and estimate the corresponding error:

?
(a)

arctan⁡1.2\arctan 1.2;

(b)

100010\sqrt[10]{1000};

(c)

1e\dfrac {1}{\sqrt{e}};

(d)

ln⁡1.25\ln 1.25.

Problem 2923

Using a suitable expansion, compute sin⁡18°\sin 18° accurate to 10−510^{-5}.

?
Problem 2924

Using a suitable expansion, compute cos⁡1°\cos 1° accurate to 10−610^{-6}.

?
Problem 2925

Using a suitable expansion, compute tan⁡9°\tan 9° accurate to 10−310^{-3}.

?
Problem 2926

Using a suitable expansion, compute ee accurate to 10−610^{-6}.

?
Problem 2927

Using a suitable expansion, compute ln⁡1.2\ln 1.2 accurate to 10−410^{-4}.

?
Problem 2928

From the equality π6=arcsin⁡12\dfrac {\pi }{6}=\arcsin \dfrac {1}{2}, find the number π\pi, accurate to 10−410^{-4}.

?
Problem 2929

Using the identity π4=arctan⁡12+arctan⁡13\dfrac {\pi }{4}=\arctan \dfrac {1}{2}+\arctan \dfrac {1}{3}, compute the number π\pi, accurate to 0.0010.001.

?
Problem 2930

Using the formula π4=4arctan⁡15−arctan⁡1239\dfrac {\pi }{4}=4\arctan \dfrac {1}{5}-\arctan \dfrac {1}{239}, find the number π\pi, accurate to 10−910^{-9}.

?
Problem 2931

Using the formula ln⁡(n+1)=ln⁡n+2(12n+1+13(2n+1)3+⋯ )\ln (n+1)=\ln n+2\left(\dfrac {1}{2n+1}+\dfrac {1}{3(2n+1)^{3}}+\cdots \right), find ln⁡2\ln 2 and ln⁡3\ln 3, accurate to 10−510^{-5}.

?
Problem 2932

Use the series expansion of the integrand to compute the value of each of the following integrals, accurate to 0.0010.001:

?
(a)

∫01e−x2 dx\displaystyle \int_{0}^{1} e^{-x^{2}}\, dx;

(b)

∫24e1x dx\displaystyle \int_{2}^{4} e^{\frac{1}{x}}\, dx;

(c)

∫02sin⁡xx dx\displaystyle \int_{0}^{2} \frac{\sin x}{x}\, dx;

(d)

∫01cos⁡x2 dx\displaystyle \int_{0}^{1} \cos x^{2}\, dx;

(e)

∫01sh⁡xx dx\displaystyle \int_{0}^{1} \frac{\operatorname {sh}x}{x}\, dx;

(f)

∫2+∞dx1+x3\displaystyle \int_{2}^{+\infty } \frac{dx}{1+x^{3}};

(g)

∫013dx1−x23\displaystyle \int_{0}^{\frac{1}{3}} \frac{dx}{\sqrt[3]{1-x^{2}}};

(h)

∫01dx1+x4\displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1+x^{4}}};

(i)

∫10100ln⁡(1+x)x dx\displaystyle \int_{10}^{100} \frac{\ln (1+x)}{x}\, dx;

(j)

∫012arctan⁡xx dx\displaystyle \int_{0}^{\frac{1}{2}} \frac{\arctan x}{x}\, dx;

(k)

∫012arcsin⁡xx dx\displaystyle \int_{0}^{\frac{1}{2}} \frac{\arcsin x}{x}\, dx;

(l)

∫01xx dx\displaystyle \int_{0}^{1} x^{x}\, dx.

Problem 2933

Find the arc length of a single half-wave of the sine curve y=sin⁡xy=\sin x (0≤x≤π0 \leq x \leq \pi), accurate to 0.010.01.

?
Problem 2934

An ellipse has semi-axes a=1a=1 and b=12b=\dfrac {1}{2}; find the arc length of the ellipse, accurate to 0.010.01.

?
Problem 2935

A wire is strung between two poles standing a distance 2l=202l=20 meters apart, and the wire hangs in the shape of a parabola. Given that the sag at the lowest point is h=40h=40 centimeters, compute the length of the wire, accurate to 11 centimeter.

?
Section 5.6Fourier Series[50/100%]
Problem 2936

Expand the function f(x)=sin⁡4xf(x)=\sin^{4}x into a Fourier series.

?
Problem 2937

What is the Fourier series of the trigonometric polynomial pn(x)=∑i=0n(αicos⁡ix+βisin⁡ix)p_{n}(x)=\displaystyle \sum_{i=0}^{n} (\alpha_{i}\cos ix+\beta_{i}\sin ix)?

?
Problem 2938

Expand the function f(x)=sgn⁡xf(x)=\operatorname {sgn}x (−π<x<π-\pi <x<\pi) into a Fourier series. Sketch the graph of the function and the graphs of several partial sums of its Fourier series. Using the expansion, find the sum of the Leibniz series ∑n=1∞(−1)n−12n−1\displaystyle \sum_{n=1}^{\infty } \dfrac {(-1)^{n-1}}{2n-1}.

?
Problem 2939

On the interval (0,2l)(0,2l), expand

f(x)={A,if 0<x<l,0,if l<x<2l, f(x)=\begin{cases} A,& \text{if } 0<x<l, \\ 0,& \text{if } l<x<2l, \end{cases}

where AA is a constant.

?
Problem 2940

On the interval (−π,π)(-\pi ,\pi ), expand f(x)=xf(x)=x.

?
Problem 2941

On the interval (0,2π)(0,2\pi ), expand f(x)=π−x2f(x)=\dfrac {\pi -x}{2}.

?
Problem 2942

On the interval (−π,π)(-\pi ,\pi ), expand f(x)=∣x∣f(x)=\left|x\right|.

?
Problem 2943

On the interval (−π,π)(-\pi ,\pi ), expand

f(x)={ax,if −π<x<0,bx,if 0<x<π, f(x)=\begin{cases} ax,& \text{if } -\pi <x<0, \\ bx,& \text{if } 0<x<\pi , \end{cases}

where aa and bb are constants.

?
Problem 2944

On the interval (−π,π)(-\pi ,\pi ), expand f(x)=π2−x2f(x)=\pi^{2}-x^{2}.

?
Problem 2945

On the interval (−π,π)(-\pi ,\pi ), expand f(x)=cos⁡axf(x)=\cos ax.

?
Problem 2946

On the interval (−π,π)(-\pi ,\pi ), expand f(x)=sin⁡axf(x)=\sin ax.

?
Problem 2947

On the interval (−π,π)(-\pi ,\pi ), expand f(x)=sh⁡axf(x)=\operatorname {sh}ax.

?
Problem 2948

On the interval (−h,h)(-h,h), expand f(x)=eaxf(x)=e^{ax}.

?
Problem 2949

On the interval (a,a+2l)(a,a+2l), expand f(x)=xf(x)=x.

?
Problem 2950

On the interval (−π,π)(-\pi ,\pi ), expand f(x)=xsin⁡xf(x)=x\sin x.

?
Problem 2951

On the interval (−π2,π2)\left(-\dfrac {\pi }{2},\dfrac {\pi }{2}\right), expand f(x)=xcos⁡xf(x)=x\cos x.

?
Problem 2952

Expand the periodic function f(x)=sgn⁡(cos⁡x)f(x)=\operatorname {sgn}(\cos x) into a Fourier series.

?
Problem 2953

Expand the periodic function f(x)=arcsin⁡(sin⁡x)f(x)=\arcsin (\sin x) into a Fourier series.

?
Problem 2954

Expand the periodic function f(x)=arcsin⁡(cos⁡x)f(x)=\arcsin (\cos x) into a Fourier series.

?
Problem 2955

Expand the periodic function f(x)=x−[x]f(x)=x-\left[x\right] into a Fourier series.

?
Problem 2956

Expand the periodic function f(x)=(x)f(x)=(x) into a Fourier series, where (x)(x) denotes the distance from xx to the nearest integer.

?
Problem 2957

Expand the periodic function f(x)=∣sin⁡x∣f(x)=\left|\sin x\right| into a Fourier series.

?
Problem 2958

Expand the periodic function f(x)=∣cos⁡x∣f(x)=\left|\cos x\right| into a Fourier series.

?
Problem 2959

Expand the periodic function f(x)=∑n=1∞αnsin⁡nxsin⁡xf(x)=\displaystyle \sum_{n=1}^{\infty } \alpha^{n}\dfrac {\sin nx}{\sin x} (∣α∣<1\left|\alpha \right|<1) into a Fourier series.

?
Problem 2960

Expand the function f(x)=sec⁡xf(x)=\sec x (−π4<x<π4-\dfrac {\pi }{4}<x<\dfrac {\pi }{4}) into a Fourier series.

?
Problem 2961

Expand the function f(x)=x2f(x)=x^{2} into a Fourier series:

?
(a)

as a cosine expansion;

(b)

as a sine expansion;

(c)

as an expansion on the interval (0,2π)(0,2\pi ).

Plot the graph of the function and the graphs of the sums of the Fourier series in cases (1), (2), and (3). Using these expansions, find the sums of the series:

∑n=1∞1n2,∑n=1∞(−1)n+1n2and∑n=1∞1(2n−1)2. \sum _{n=1}^{\infty } \frac{1}{n^{2}}, \quad \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{n^{2}} \quad \text{and} \quad \sum _{n=1}^{\infty } \frac{1}{(2n-1)^{2}}.
Problem 2962

From the expansion

x=2∑n=1∞(−1)n+1sin⁡nxn(−π<x<π), x=2\sum _{n=1}^{\infty } (-1)^{n+1}\frac{\sin nx}{n} \quad (-\pi <x<\pi ),

using term-by-term integration, find the Fourier series of the functions x2x^{2}, x3x^{3}, and x4x^{4} on the interval (−π,π)(-\pi ,\pi ).

?
Problem 2963

Write out the Lyapunov equation for the function

f(x)={1,when∣x∣<a,0,whena<∣x∣<π f(x)= \begin{cases} 1,& \text{when} \left|x\right|<a,\\ 0,& \text{when} a<\left|x\right|<\pi \end{cases}

Using the Lyapunov equation, find the sums of the following series:

∑n=1∞sin⁡2nan2and∑n=1∞cos⁡2nan2. \sum _{n=1}^{\infty } \frac{\sin ^{2}na}{n^{2}} \quad \text{and} \quad \sum _{n=1}^{\infty } \frac{\cos ^{2}na}{n^{2}}.
?
Problem 2964

Expand the function

f(x)={x,if0⩽x⩽1,1,if1<x<2,3−x,if2⩽x⩽3 f(x)= \begin{cases} x,& \text{if} 0 \leqslant x \leqslant 1,\\ 1,& \text{if} 1<x<2,\\ 3-x,& \text{if} 2 \leqslant x \leqslant 3 \end{cases}

into a Fourier series.

?
Problem 2965

Using the formulas

cos⁡x=12(t+tˉ),sin⁡x=12i(t−tˉ), \cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where t=eixt=e^{ix} and tˉ=e−ix\bar{t}=e^{-ix}, expand the following function into a Fourier series: cos⁡2mx\cos^{2m}x (mm a positive integer).

?
Problem 2966

Using the formulas

cos⁡x=12(t+tˉ),sin⁡x=12i(t−tˉ), \cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where t=eixt=e^{ix} and tˉ=e−ix\bar{t}=e^{-ix}, expand the following function into a Fourier series:

qsin⁡x1−2qcos⁡x+q2(∣q∣<1). \frac{q\sin x}{1-2q\cos x+q^{2}} \quad (\left|q\right|<1).
?
Problem 2967

Using the formulas

cos⁡x=12(t+tˉ),sin⁡x=12i(t−tˉ), \cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where t=eixt=e^{ix} and tˉ=e−ix\bar{t}=e^{-ix}, expand the following function into a Fourier series:

1−q21−2qcos⁡x+q2(∣q∣<1). \frac{1-q^{2}}{1-2q\cos x+q^{2}} \quad (\left|q\right|<1).
?
Problem 2968

Using the formulas

cos⁡x=12(t+tˉ),sin⁡x=12i(t−tˉ), \cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where t=eixt=e^{ix} and tˉ=e−ix\bar{t}=e^{-ix}, expand the following function into a Fourier series:

1−qcos⁡x1−2qcos⁡x+q2(∣q∣<1). \frac{1-q\cos x}{1-2q\cos x+q^{2}} \quad (\left|q\right|<1).
?
Problem 2969

Using the formulas

cos⁡x=12(t+tˉ),sin⁡x=12i(t−tˉ), \cos x=\frac{1}{2}(t+\bar{t}), \quad \sin x=\frac{1}{2i}(t-\bar{t}),

where t=eixt=e^{ix} and tˉ=e−ix\bar{t}=e^{-ix}, expand the following function into a Fourier series:

ln⁡(1−2qcos⁡x+q2)(∣q∣<1). \ln (1-2q\cos x+q^{2}) \quad (\left|q\right|<1).
?
Problem 2970

Expand the following unbounded periodic function into a Fourier series:

f(x)=ln⁡∣sin⁡x2∣. f(x)=\ln \left|\sin \frac{x}{2}\right|.
?
Problem 2971

Expand the following unbounded periodic function into a Fourier series:

f(x)=ln⁡∣cos⁡x2∣. f(x)=\ln \left|\cos \frac{x}{2}\right|.
?
Problem 2972

Expand the following unbounded periodic function into a Fourier series:

f(x)=ln⁡∣tg⁡x2∣. f(x)=\ln \left|\operatorname {tg}\frac{x}{2}\right|.
?
Problem 2973

Expand the function

f(x)=∫0xln⁡∣ctg⁡t2∣ dt(−π⩽x⩽π) f(x)=\int _{0}^{x} \ln \sqrt{\left|\operatorname {ctg}\frac{t}{2}\right|}\, dt \quad (-\pi \leqslant x \leqslant \pi )

into a Fourier series.

?
Problem 2974

The functions

x=x(s),y=y(s)(0⩽s⩽4a) x=x(s), \quad y=y(s) \quad (0 \leqslant s \leqslant 4a)

are the parametric equations of the boundary of the square: 0<x<a0<x<a, 0<y<a0<y<a, where ss is the arc length measured counterclockwise starting from the point O(0,0)O(0,0). Expand this function into a Fourier series.

?
Problem 2975

How should the integrable function f(x)f(x), given on the interval (0,π2)\left(0,\dfrac {\pi }{2}\right), be extended to the interval (−π,π)(-\pi ,\pi ) so that its expansion into a Fourier series has the form

f(x)=∑n=1∞ancos⁡(2n−1)x(−π<x<π)? f(x)=\sum _{n=1}^{\infty } a_{n}\cos (2n-1)x \quad (-\pi <x<\pi )?
?
Problem 2976

How should the integrable function f(x)f(x), given on the interval (0,π2)\left(0,\dfrac {\pi }{2}\right), be extended to the interval (−π,π)(-\pi ,\pi ) so that its expansion into a Fourier series has the form

f(x)=∑n=1∞bnsin⁡(2n−1)x(−π<x<π)? f(x)=\sum _{n=1}^{\infty } b_{n}\sin (2n-1)x \quad (-\pi <x<\pi )?
?
Problem 2977

On the interval (0,π2)\left(0,\dfrac {\pi }{2}\right), expand the function

f(x)=x(π2−x) f(x)=x\left(\frac{\pi }{2}-x\right)

as follows:

?
(a)

in cosines of odd multiples of the angle;

(b)

in sines of odd multiples of the angle.

Sketch the graphs of the sums of the Fourier series in cases (1) and (2).

Problem 2978

Let f(x)f(x) be an anti-periodic function with period π\pi, that is,

f(x+π)=−f(x). f(x+\pi )=-f(x).

What special property does the Fourier series of this function on the interval (−π,π)(-\pi ,\pi ) have?

?
Problem 2979

Let f(x+π)=f(x)f(x+\pi )=f(x). What special property does the Fourier series of the function f(x)f(x) on the interval (−π,π)(-\pi ,\pi ) have?

?
Problem 2980

A function y=f(x)y=f(x) with period 2π2\pi has a graph that is:

?
(a)

symmetric about the points (0,0)(0,0), (±π2,0)\left(\pm \dfrac {\pi }{2},0\right) as centers;

(b)

symmetric about the origin as a center and about x=±π2x=\pm \dfrac {\pi }{2} as axes of symmetry,

What special properties do its Fourier coefficients ana_{n}, bnb_{n} (n=1,2,3,⋯n=1,2,3,\cdots) have in each case?

Problem 2981

If the functions satisfy φ(−x)=ψ(x)\varphi (-x)=\psi (x), what relation holds between the Fourier coefficients an,bna_{n},b_{n} of φ(x)\varphi (x) and αn,βn\alpha_{n},\beta_{n} of ψ(x)\psi (x) (n=0,1,2,⋯n=0,1,2,\cdots)?

?
Problem 2982

If the functions satisfy φ(−x)=−ψ(x)\varphi (-x)=-\psi (x), what relation holds between the Fourier coefficients an,bna_{n},b_{n} of φ(x)\varphi (x) and αn,βn\alpha_{n},\beta_{n} of ψ(x)\psi (x) (n=0,1,2,⋯n=0,1,2,\cdots)?

?
Problem 2983

Given that the integrable function f(x)f(x) with period 2π2\pi has Fourier coefficients an,bna_{n},b_{n} (n=0,1,2,⋯n=0,1,2,\cdots), compute the Fourier coefficients an‾,bn‾\overline{a_{n}},\overline{b_{n}} (n=0,1,2,⋯n=0,1,2,\cdots) of the "shifted" function f(x+h)f(x+h) (h=h= constant).

?
Problem 2984

Given that the integrable function f(x)f(x) with period 2π2\pi has Fourier coefficients an,bna_{n},b_{n} (n=0,1,2,⋯n=0,1,2,\cdots), compute the Fourier coefficients An,BnA_{n},B_{n} (n=0,1,2,⋯n=0,1,2,\cdots) of the Steklov function

fh(x)=12h∫x−hx+hf(ξ) dξ f_{h}(x)=\frac{1}{2h}\int _{x-h}^{x+h} f(\xi )\, d\xi
?
Problem 2985

Let f(x)f(x) be a continuous function with period 2π2\pi, and let an,bna_{n},b_{n} (n=0,1,2,⋯n=0,1,2,\cdots) be its Fourier coefficients. Find the Fourier coefficients An,BnA_{n},B_{n} (n=0,1,2,⋯n=0,1,2,\cdots) of the convolution function

F(x)=1π∫−ππf(t)f(x+t) dt F(x)=\frac{1}{\pi }\int _{-\pi }^{\pi } f(t)f(x+t)\, dt

Using the result obtained, derive Lyapunov's equality.

?
Section 5.7Methods of Summing Series[48/100%]
Problem 2986

Find the sum of the following series:

11⋅3+13⋅5+15⋅7+⋯ . \frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots .
?
Problem 2987

Find the sum of the following series:

11⋅2⋅3+12⋅3⋅4+13⋅4⋅5+⋯ . \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}+\cdots .
?
Problem 2988

Find the sum of the following series:

11⋅2−12⋅3+13⋅4−14⋅5+⋯ . \frac{1}{1 \cdot 2}-\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}-\frac{1}{4 \cdot 5}+\cdots .
?
Problem 2989

Find the sum of the following series:

∑n=1∞n(n+1)(n+2)(n+3). \sum _{n=1}^{\infty } \frac{n}{(n+1)(n+2)(n+3)}.
?
Problem 2990

Find the sum of the following series:

∑n=1∞1n(n+m)(m is a natural number). \sum _{n=1}^{\infty } \frac{1}{n(n+m)} \quad (m \text{ is a natural number}).
?
Problem 2991

Find the sum of the following series:

11⋅2⋅3+13⋅4⋅5+15⋅6⋅7+⋯ . \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{3 \cdot 4 \cdot 5}+\frac{1}{5 \cdot 6 \cdot 7}+\cdots .
?
Problem 2992

Find the sum of the following series:

∑n=2∞1n2−1. \sum _{n=2}^{\infty } \frac{1}{n^{2}-1}.
?
Problem 2993

Find the sum of the following series:

∑n=1∞2n+1n2(n+1)2. \sum _{n=1}^{\infty } \frac{2n+1}{n^{2}(n+1)^{2}}.
?
Problem 2994

Find the sum of the following series:

∑n=1∞1n(2n+1). \sum _{n=1}^{\infty } \frac{1}{n(2n+1)}.
?
Problem 2995

Find the sum of the following series:

∑n=1∞n2n!. \sum _{n=1}^{\infty } \frac{n^{2}}{n!}.
?
Problem 2996

Find the sum of the following series:

∑n=0∞2n(n+1)n!. \sum _{n=0}^{\infty } \frac{2^{n}(n+1)}{n!}.
?
Problem 2997

Find the sum of the following series:

∑n=1∞1n2(n+1)2. \sum _{n=1}^{\infty } \frac{1}{n^{2}(n+1)^{2}}.
?
Problem 2998

Find the sum of the following series:

∑n=1∞1n2(n+1)2(n+2)2. \sum _{n=1}^{\infty } \frac{1}{n^{2}(n+1)^{2}(n+2)^{2}}.
?
Problem 2999

Find the sum of the following series:

∑n=0∞(−1)nn(2n+1)!. \sum _{n=0}^{\infty } \frac{(-1)^{n}n}{(2n+1)!}.
?
Problem 3000

Find the sum of the following series:

∑n=2∞(−1)nn2+n−2. \sum _{n=2}^{\infty } \frac{(-1)^{n}}{n^{2}+n-2}.
?
Problem 3001

Let P(x)=α0+α1x+⋯+αmxmP(x)=\alpha_{0}+\alpha_{1}x+\cdots +\alpha_{m}x^{m}. Find the sum of the series

∑n=0∞P(n)n!xn \sum _{n=0}^{\infty } \frac{P(n)}{n!}x^{n}
?
Problem 3002

Find the sum of the following series:

∑n=0∞n2+12nn!xn. \sum _{n=0}^{\infty } \frac{n^{2}+1}{2^{n}n!}x^{n}.
?
Problem 3003

Find the sum of the following series:

∑n=0∞(−1)nn3(n+1)!xn. \sum _{n=0}^{\infty } \frac{(-1)^{n}n^{3}}{(n+1)!}x^{n}.
?
Problem 3004

Find the sum of the following series:

∑n=0∞(−1)n(2n2+1)(2n)!x2n. \sum _{n=0}^{\infty } \frac{(-1)^{n}(2n^{2}+1)}{(2n)!}x^{2n}.
?
Problem 3005

Find the sum of the following series:

∑n=0∞n2xn(2n+1)!. \sum _{n=0}^{\infty } \frac{n^{2}x^{n}}{(2n+1)!}.
?
Problem 3006

Using term-by-term differentiation, find the sum of the following series:

∑n=1∞xnn. \sum _{n=1}^{\infty } \frac{x^{n}}{n}.
?
Problem 3007

Using term-by-term differentiation, find the sum of the following series:

∑n=1∞(−1)n−1x2nn(2n−1). \sum _{n=1}^{\infty } \frac{(-1)^{n-1}x^{2n}}{n(2n-1)}.
?
Problem 3008

Using term-by-term differentiation, find the sum of the following series:

∑n=0∞x4n+14n+1. \sum _{n=0}^{\infty } \frac{x^{4n+1}}{4n+1}.
?
Problem 3009

Find the sum of the following series:

∑n=1∞a(a+d)⋯[a+(n−1)d]d⋅2d⋯ndxn(d>0). \sum _{n=1}^{\infty } \frac{a(a+d)\cdots \left[a+(n-1)d\right]}{d \cdot 2d\cdots nd}x^{n} \quad (d>0).
?
Problem 3010

Find the sum of the following series:

13(x2)+1⋅43⋅6(x2)2+1⋅4⋅73⋅6⋅9(x2)3+⋯ . \frac{1}{3}\left(\frac{x}{2}\right)+\frac{1 \cdot 4}{3 \cdot 6}\left(\frac{x}{2}\right)^{2}+\frac{1 \cdot 4 \cdot 7}{3 \cdot 6 \cdot 9}\left(\frac{x}{2}\right)^{3}+\cdots .
?
Problem 3011

Using term-by-term integration, find the sum of the following series:

∑n=1∞n2xn−1. \sum _{n=1}^{\infty } n^{2}x^{n-1}.
?
Problem 3012

Using term-by-term integration, find the sum of the following series:

∑n=1∞n(n+2)xn. \sum _{n=1}^{\infty } n(n+2)x^{n}.
?
Problem 3013

Using term-by-term integration, find the sum of the following series:

∑n=0∞(2n+1)x2nn!. \sum _{n=0}^{\infty } \frac{(2n+1)x^{2n}}{n!}.
?
Problem 3014

Use Abel's method to find the sum of the following series:

1−14+17−110+⋯ . 1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots .
?
Problem 3015

Use Abel's method to find the sum of the following series:

1−13+15−17+⋯ . 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots .
?
Problem 3016

Use Abel's method to find the sum of the following series:

1−12+1⋅32⋅4−1⋅3⋅52⋅4⋅6+⋯ . 1-\frac{1}{2}+\frac{1 \cdot 3}{2 \cdot 4}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}+\cdots .
?
Problem 3017

Use Abel's method to find the sum of the following series:

1+12⋅13+1⋅32⋅4⋅15+⋯ . 1+\frac{1}{2} \cdot \frac{1}{3}+\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{5}+\cdots .
?
Problem 3018

Find the sum of the following trigonometric series:

∑n=1∞sin⁡nxn. \sum _{n=1}^{\infty } \frac{\sin nx}{n}.
?
Problem 3019

Find the sum of the following trigonometric series:

∑n=1∞cos⁡nxn. \sum _{n=1}^{\infty } \frac{\cos nx}{n}.
?
Problem 3020

Find the sum of the following trigonometric series:

∑n=1∞sin⁡nαsin⁡nxn. \sum _{n=1}^{\infty } \frac{\sin n\alpha \sin nx}{n}.
?
Problem 3021

Find the sum of the following trigonometric series:

∑n=1∞sin⁡2nαsin⁡nxn(0<α<π2). \sum _{n=1}^{\infty } \frac{\sin ^{2} n\alpha \sin nx}{n} \quad \left(0<\alpha <\frac{\pi }{2}\right).
?
Problem 3022

Find the sum of the following trigonometric series:

∑n=1∞sin⁡(2n−1)x2n−1. \sum _{n=1}^{\infty } \frac{\sin (2n-1)x}{2n-1}.
?
Problem 3023

Find the sum of the following trigonometric series:

∑n=2∞(−1)ncos⁡nxn2−1. \sum _{n=2}^{\infty } (-1)^{n} \frac{\cos nx}{n^{2}-1}.
?
Problem 3024

Find the sum of the following trigonometric series:

∑n=1∞cos⁡(2n−1)x(2n−1)2. \sum _{n=1}^{\infty } \frac{\cos (2n-1)x}{(2n-1)^{2}}.
?
Problem 3025

Find the sum of the following trigonometric series:

∑n=1∞(−1)n−1sin⁡nxn(n+1). \sum _{n=1}^{\infty } (-1)^{n-1} \frac{\sin nx}{n(n+1)}.
?
Problem 3026

Find the sum of the following trigonometric series:

∑n=0∞cos⁡nxn!. \sum _{n=0}^{\infty } \frac{\cos nx}{n!}.
?
Problem 3027

Plot the curve

∑n=1∞sin⁡nx⋅sin⁡nyn2=0 \sum _{n=1}^{\infty } \frac{\sin nx \cdot \sin ny}{n^{2}}=0
?
Problem 3028

Find the sum of the following series:

∑n=1∞[(n−1)!]2(2n)!(2x)2n. \sum _{n=1}^{\infty } \frac{\left[(n-1)!\right]^{2}}{(2n)!}(2x)^{2n}.
?
Problem 3029

Find the sum of the following series:

∑n=0∞(n!)2(2n)!xn. \sum _{n=0}^{\infty } \frac{(n!)^{2}}{(2n)!}x^{n}.
?
Problem 3030

Find the sum of the following series:

1!x+1+2!(x+1)(x+2)+3!(x+1)(x+2)(x+3)+⋯ . \frac{1!}{x+1}+\frac{2!}{(x+1)(x+2)}+\frac{3!}{(x+1)(x+2)(x+3)}+\cdots .
?
Problem 3031

Discuss the series

a1a2+x+a1a2+x⋅a2a3+x+⋯ \frac{a_{1}}{a_{2}+x}+\frac{a_{1}}{a_{2}+x} \cdot \frac{a_{2}}{a_{3}+x}+\cdots

for its convergence and sum, under the condition x>0x>0, an>0a_{n}>0 (n=1,2,⋯n=1,2,\cdots), given that the series ∑n=1∞1an\displaystyle \sum_{n=1}^{\infty } \frac{1}{a_{n}} diverges.

?
Problem 3032

Find the sum of the following series:

x1−x2+x21−x4+x41−x8+⋯ , \frac{x}{1-x^{2}}+\frac{x^{2}}{1-x^{4}}+\frac{x^{4}}{1-x^{8}}+\cdots ,

if (a) ∣x∣<1\left|x\right|<1; (б) ∣x∣>1\left|x\right|>1.

?
(a)
(b)
Problem 3033

Find the sum of the following series:

∑n=1∞xn+1(1−xn)(1−xn+1), \sum _{n=1}^{\infty } \frac{x^{n+1}}{(1-x^{n})(1-x^{n+1})},

if (a) ∣x∣<1\left|x\right|<1; (б) ∣x∣>1\left|x\right|>1.

?
(a)
(b)
Section 5.8Evaluating Definite Integrals Using Series[17/100%]
Problem 3034

Using the series expansion of the integrand, compute the following integral:

∫01ln⁡11−x dx. \int _{0}^{1} \ln \frac{1}{1-x}\, dx.
?
Problem 3035

Using the series expansion of the integrand, compute the following integral:

∫01ln⁡(x+1+x2)x dx. \int _{0}^{1} \frac{\ln (x+\sqrt{1+x^{2}})}{x}\, dx.
?
Problem 3036

Using the series expansion of the integrand, compute the following integral:

∫01ln⁡(1+x)x dx. \int _{0}^{1} \frac{\ln (1+x)}{x}\, dx.
?
Problem 3037

Using the series expansion of the integrand, compute the following integral:

∫01xp−1ln⁡(1−xq) dx(p>0,q>0). \int _{0}^{1} x^{p-1}\ln (1-x^{q})\, dx \quad (p>0,q>0).
?
Problem 3038

Using the series expansion of the integrand, compute the following integral:

∫01ln⁡x⋅ln⁡(1−x) dx. \int _{0}^{1} \ln x \cdot \ln (1-x)\, dx.
?
Problem 3039

Using the series expansion of the integrand, compute the following integral:

∫0+∞x dxe2πx−1. \int _{0}^{+\infty } \frac{x\, dx}{e^{2\pi x}-1}.
?
Problem 3040

Using the series expansion of the integrand, compute the following integral:

∫0+∞x dxex+1. \int _{0}^{+\infty } \frac{x\, dx}{e^{x}+1}.
?
Problem 3041

Expand the complete elliptic integral of the first kind in positive integer powers of the modulus kk (0⩽k<10 \leqslant k<1):

F(k)=∫0π2dφ1−k2sin⁡2φ. F(k)=\int _{0}^{\frac{\pi }{2}} \frac{d\varphi }{\sqrt{1-k^{2}\sin ^{2}\varphi }}.
?
Problem 3042

Expand the complete elliptic integral of the second kind in positive integer powers of the modulus kk (0⩽k<10 \leqslant k<1):

E(k)=∫0π21−k2sin⁡2φ dφ. E(k)=\int _{0}^{\frac{\pi }{2}} \sqrt{1-k^{2}\sin ^{2}\varphi }\, d\varphi .
?
Problem 3043

Express the arc length of the following ellipse as a series expanded in positive integer powers of the eccentricity:

x=acos⁡t,x=bsin⁡t(0⩽t⩽2π). x=a\cos t,\quad x=b\sin t \quad (0 \leqslant t \leqslant 2\pi ).
?
Problem 3044

Prove the following identity:

∫01dxxx=∑n=1∞1nn. \int _{0}^{1} \frac{dx}{x^{x}}=\sum _{n=1}^{\infty } \frac{1}{n^{n}}.
?
Problem 3045

Prove the following identity:

∫0+∞e−x2sin⁡ax dx=12∑n=0∞(−1)nn!(2n+1)!a2n+1. \int _{0}^{+\infty } e^{-x^{2}}\sin ax\, dx=\frac{1}{2}\sum _{n=0}^{\infty } \frac{(-1)^{n}n!}{(2n+1)!}a^{2n+1}.
?
Problem 3046

Prove the following identity:

∫02πecos⁡xcos⁡(sin⁡x)cos⁡nx dx=πn!(n=0,1,2,⋯ ). \int _{0}^{2\pi } e^{\cos x}\cos (\sin x)\cos nx\, dx=\frac{\pi }{n!} \quad (n=0,1,2,\cdots ).
?
Problem 3047

Find:

∫02πeacos⁡xcos⁡(asin⁡x−nx) dx(n is a natural number). \int _{0}^{2\pi } e^{a\cos x}\cos (a\sin x-nx)\, dx \quad (n\text{ is a natural number}).
?
Problem 3048

Find:

∫0πxsin⁡x1−2acos⁡x+a2 dx. \int _{0}^{\pi } \frac{x\sin x}{1-2a\cos x+a^{2}}\, dx.
?
Problem 3049

Find:

∫0πln⁡(1−2acos⁡x+a2) dx. \int _{0}^{\pi } \ln (1-2a\cos x+a^{2})\, dx.
?
Problem 3050

Prove the formula:

∫0∞e−xa+x dx=1a−1!a2+2!a3−⋯+(−1)n−1(n−1)!an+(−1)nθnn!an+1, \begin{align} \int _{0}^{\infty } \frac{e^{-x}}{a+x}\, dx & =\frac{1}{a}-\frac{1!}{a^{2}}+\frac{2!}{a^{3}}-\cdots +(-1)^{n-1}\frac{(n-1)!}{a^{n}} \\ & \quad +(-1)^{n}\frac{\theta _{n}n!}{a^{n+1}}, \tag {1} \end{align}

where a>0a>0 and 0<θn<10<\theta_{n}<1.

If two terms in formula (1) are taken to represent the integral

∫0+∞e−x100+x dx, \int _{0}^{+\infty } \frac{e^{-x}}{100+x}\, dx,

what is the precision of this approximation?

?
Section 5.9Infinite Products[60/100%]
Problem 3051

Prove the following identity:

∏n=2∞(1−1n2)=12. \prod _{n=2}^{\infty } \left(1-\frac{1}{n^{2}}\right)=\frac{1}{2}.
?
Problem 3052

Prove the following identity:

∏n=2∞n3−1n3+1=23. \prod _{n=2}^{\infty } \frac{n^{3}-1}{n^{3}+1}=\frac{2}{3}.
?
Problem 3053

Prove the following identity:

∏n=2∞(1−2n(n+1))=13. \prod _{n=2}^{\infty } \left(1-\frac{2}{n(n+1)}\right)=\frac{1}{3}.
?
Problem 3054

Prove the following identity:

∏n=0∞[1+(12)2n]=2. \prod _{n=0}^{\infty } \left[1+\left(\frac{1}{2}\right)^{2^{n}}\right]=2.
?
Problem 3055

Prove the following identity:

∏n=1∞cos⁡π2n+1=2π. \prod _{n=1}^{\infty } \cos \frac{\pi }{2^{n+1}}=\frac{2}{\pi }.
?
Problem 3056

Prove the following identity:

∏n=1∞cos⁡x2n=sin⁡xx. \prod _{n=1}^{\infty } \cos \frac{x}{2^{n}}=\frac{\sin x}{x}.
?
Problem 3057

Prove the following identity:

∏n=1∞ch⁡x2n=sh⁡xx. \prod _{n=1}^{\infty } \ch \frac{x}{2^{n}}=\frac{\sh x}{x}.
?
Problem 3058

Prove the following identity:

∏n=0∞(1+x2n)=11−x(∣x∣<1). \prod _{n=0}^{\infty } (1+x^{2^{n}})=\frac{1}{1-x} \quad (\left|x\right|<1).
?
Problem 3059

Prove the following identity:

π2=22⋅22+2⋅22+2+2⋯ . \frac{\pi }{2}=\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{2+\sqrt{2}}} \cdot \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdots .
?
Problem 3060

Prove the following identity:

∏n=1∞3n3n−1⋅3n3n+1=2π33. \prod _{n=1}^{\infty } \frac{3n}{3n-1} \cdot \frac{3n}{3n+1}=\frac{2\pi }{3\sqrt{3}}.
?
Problem 3061

Prove the convergence of the following infinite product and find its value:

∏n=3∞n2−4n2−1. \prod _{n=3}^{\infty } \frac{n^{2}-4}{n^{2}-1}.
?
Problem 3062

Prove the convergence of the following infinite product and find its value:

∏n=1∞(1+1n(n+2)). \prod _{n=1}^{\infty } \left(1+\frac{1}{n(n+2)}\right).
?
Problem 3063

Prove the convergence of the following infinite product and find its value:

∏n=1∞(2n+1)(2n+7)(2n+3)(2n+5). \prod _{n=1}^{\infty } \frac{(2n+1)(2n+7)}{(2n+3)(2n+5)}.
?
Problem 3064

Prove the convergence of the following infinite product and find its value:

∏n=1∞a(−1)nn(a>0). \prod _{n=1}^{\infty } a^{\frac{(-1)^{n}}{n}} \quad (a>0).
?
Problem 3065

Can the convergence of the following products be deduced from the convergence of the products ∏n=1∞pn\displaystyle \prod_{n=1}^{\infty } p_{n} and ∏n=1∞qn\displaystyle \prod_{n=1}^{\infty } q_{n}?

  1. ∏n=1∞(pn+qn)\displaystyle \prod_{n=1}^{\infty } (p_{n}+q_{n});

  2. ∏n=1∞pn2\displaystyle \prod_{n=1}^{\infty } p_{n}^{2};

  3. ∏n=1∞pnqn\displaystyle \prod_{n=1}^{\infty } p_{n}q_{n};

  4. ∏n=1∞pnqn\displaystyle \prod_{n=1}^{\infty } \frac{p_{n}}{q_{n}}.

?
(a)
(b)
(c)
(d)
Problem 3066

Investigate the convergence of the following infinite product:

∏n=1∞1n. \prod _{n=1}^{\infty } \frac{1}{n}.
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Problem 3067

Investigate the convergence of the following infinite product:

∏n=1∞(n+1)2n(n+2). \prod _{n=1}^{\infty } \frac{(n+1)^{2}}{n(n+2)}.
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Problem 3068

Investigate the convergence of the following infinite product:

∏n=1∞(1+1np). \prod _{n=1}^{\infty } \left(1+\frac{1}{n^{p}}\right).
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Problem 3069

Investigate the convergence of the following infinite product:

∏n=2∞(1−1n). \prod _{n=2}^{\infty } \left(1-\frac{1}{n}\right).
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Problem 3070

Investigate the convergence of the following infinite product:

∏n=2∞(1−2n2+1)p∗). \prod _{n=2}^{\infty } \left(1-\frac{2}{n^{2}+1}\right)^{p} {}^{*)}.
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Problem 3071

Investigate the convergence of the following infinite product:

∏n=n0∞n2+a1n+b1n2+an+b, \prod _{n=n_{0}}^{\infty } \frac{n^{2}+a_{1}n+b_{1}}{n^{2}+an+b},

where n2+an+b>0n^{2}+an+b>0 for n≥n0n \geq n_{0}.

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Problem 3072

Investigate the convergence of the following infinite product:

∏n=n0∞(n−a1)(n−a2)⋯(n−ap)(n−b1)(n−b2)⋯(n−bp), \prod _{n=n_{0}}^{\infty } \frac{(n-a_{1})(n-a_{2})\cdots (n-a_{p})}{(n-b_{1})(n-b_{2})\cdots (n-b_{p})},

where n0>bin_{0}>b_{i} (i=1,2,⋯ ,pi=1,2,\cdots ,p).

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Problem 3073

Investigate the convergence of the following infinite product:

∏n=0∞n+1n+2. \prod _{n=0}^{\infty } \sqrt{\frac{n+1}{n+2}}.
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Problem 3074

Investigate the convergence of the following infinite product:

∏n=1∞nn2+1. \prod _{n=1}^{\infty } \frac{n}{\sqrt{n^{2}+1}}.
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Problem 3075

Investigate the convergence of the following infinite product:

∏n=1∞1+1nn. \prod _{n=1}^{\infty } \sqrt[n]{1+\frac{1}{n}}.
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Problem 3076

Investigate the convergence of the following infinite product:

∏n=1∞nn2. \prod _{n=1}^{\infty } \sqrt[n^{2}]{n}.
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Problem 3077

Investigate the convergence of the following infinite product:

∏n=1∞(1+xn)e−xn. \prod _{n=1}^{\infty } \left(1+\frac{x}{n}\right)e^{-\frac{x}{n}}.
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Problem 3078

Investigate the convergence of the following infinite product:

∏n=1∞(1−xc+n)exn, \prod _{n=1}^{\infty } \left(1-\frac{x}{c+n}\right)e^{\frac{x}{n}},

where c>0c>0.

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Problem 3079

Investigate the convergence of the following infinite product:

∏n=1∞(1−xn). \prod _{n=1}^{\infty } (1-x^{n}).
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Problem 3080

Investigate the convergence of the following infinite product:

∏n=1∞(1+xn2n). \prod _{n=1}^{\infty } \left(1+\frac{x^{n}}{2^{n}}\right).
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Problem 3081

Investigate the convergence of the following infinite product:

∏n=1∞[1+(1+1n)n2xn]. \prod _{n=1}^{\infty } \left[1+\frac{\left(1+\dfrac {1}{n}\right)^{n^{2}}}{x^{n}}\right].
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Problem 3082

Investigate the convergence of the following infinite product:

∏n=1∞(1−xn)exn+x22n. \prod _{n=1}^{\infty } \left(1-\frac{x}{\sqrt{n}}\right)e^{\frac{x}{\sqrt{n}}+\frac{x^{2}}{2n}}.
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Problem 3083

Investigate the convergence of the following infinite product:

∏n=1∞(1+xnnp)cos⁡xnnq. \prod _{n=1}^{\infty } \left(1+\frac{x^{n}}{n^{p}}\right)\cos \frac{x^{n}}{n^{q}}.
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Problem 3084

Investigate the convergence of the following infinite product:

∏n=1∞(sin⁡xnxn)p. \prod _{n=1}^{\infty } \left(\frac{\sin \dfrac {x}{n}}{\dfrac {x}{n}}\right)^{p}.
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Problem 3085

Investigate the convergence of the following infinite product:

∏n=1∞ln⁡(n+x)−ln⁡nn. \prod _{n=1}^{\infty } \sqrt[n]{\ln (n+x)-\ln n}.
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Problem 3086

Prove: if the series ∑n=1∞xn2\displaystyle \sum_{n=1}^{\infty } x_{n}^{2} converges, then the product ∏n=1∞cos⁡xn\displaystyle \prod_{n=1}^{\infty } \cos x_{n} converges.

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Problem 3087

Prove: if the series ∑n=1∞αn\displaystyle \sum_{n=1}^{\infty } \alpha_{n} converges absolutely, then the product ∏n=1∞tan⁡(π4+αn)\displaystyle \prod_{n=1}^{\infty } \tan \left(\frac{\pi }{4}+\alpha_{n}\right) (∣αn∣<π4\left|\alpha_{n}\right|<\dfrac {\pi }{4}) converges.

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Problem 3088

Investigate the absolute convergence and conditional convergence of the following infinite product:

∏n=1∞(1+(−1)n+1n). \prod _{n=1}^{\infty } \left(1+\frac{(-1)^{n+1}}{n}\right).
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Problem 3089

Investigate the absolute convergence and conditional convergence of the following infinite product:

∏n=1∞(1+(−1)n+1n). \prod _{n=1}^{\infty } \left(1+\frac{(-1)^{n+1}}{\sqrt{n}}\right).
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Problem 3090

Investigate the absolute convergence and conditional convergence of the following infinite product:

∏n=1∞(1+(−1)n+1np). \prod _{n=1}^{\infty } \left(1+\frac{(-1)^{n+1}}{n^{p}}\right).
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Problem 3091

Investigate the absolute and conditional convergence of the following infinite product:

∏n=2∞(1+(−1)nln⁡n). \prod _{n=2}^{\infty } \left(1+\frac{(-1)^{n}}{\ln n}\right).
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Problem 3092

Investigate the absolute and conditional convergence of the following infinite product:

∏n=2∞nn+(−1)n. \prod _{n=2}^{\infty } \frac{\sqrt{n}}{\sqrt{n}+(-1)^{n}}.
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Problem 3093

Investigate the absolute and conditional convergence of the following infinite product:

∏n=1∞n(−1)n. \prod _{n=1}^{\infty } n^{(-1)^{n}}.
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Problem 3094

Investigate the absolute and conditional convergence of the following infinite product:

∏n=1∞n(−1)nn. \prod _{n=1}^{\infty } \sqrt[n]{n^{(-1)^{n}}}.
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Problem 3095

Investigate the absolute and conditional convergence of the following infinite product:

∏n=1∞[1+(−1)n(n−1)2n]. \prod _{n=1}^{\infty } \left[1+\frac{(-1)^{\frac{n(n-1)}{2}}}{n}\right].
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Problem 3096

Investigate the absolute and conditional convergence of the following infinite product:

(1+11)(1−13)(1−15)×(1+12)(1−17)(1−19)(1+13)⋯ . \begin{align} & \left(1+\frac{1}{\sqrt{1}}\right)\left(1-\frac{1}{\sqrt{3}}\right)\left(1-\frac{1}{\sqrt{5}}\right) \\ & \quad \times \left(1+\frac{1}{\sqrt{2}}\right)\left(1-\frac{1}{\sqrt{7}}\right)\left(1-\frac{1}{\sqrt{9}}\right)\left(1+\frac{1}{\sqrt{3}}\right)\cdots . \end{align}
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Problem 3097

Investigate the absolute and conditional convergence of the following infinite product:

(1+11a)(1−12a)2(1+13a)(1+14a)(1−15a)2×(1+16a)⋯ . \begin{align} & \left(1+\frac{1}{1^{a}}\right)\left(1-\frac{1}{2^{a}}\right)^{2}\left(1+\frac{1}{3^{a}}\right)\left(1+\frac{1}{4^{a}}\right)\left(1-\frac{1}{5^{a}}\right)^{2} \\ & \quad \times \left(1+\frac{1}{6^{a}}\right)\cdots . \end{align}
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Problem 3098

Prove that although the series

(12+12)+(−12)+(13+13)+(−13)+⋯(1) \left(\frac{1}{\sqrt{2}}+\frac{1}{2}\right)+\left(-\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{3}}+\frac{1}{3}\right)+\left(-\frac{1}{\sqrt{3}}\right)+\cdots \tag {1}

diverges, the product

(1+12+12)(1−12)(1+13+13)(1−13)⋯(2) \left(1+\frac{1}{\sqrt{2}}+\frac{1}{2}\right)\left(1-\frac{1}{\sqrt{2}}\right)\left(1+\frac{1}{\sqrt{3}}+\frac{1}{3}\right)\left(1-\frac{1}{\sqrt{3}}\right)\cdots \tag {2}

converges.

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Problem 3099

Prove that although the two series

∑n=1∞αnand∑n=1∞αn2 \sum _{n=1}^{\infty } \alpha _{n} \quad \text{and} \quad \sum _{n=1}^{\infty } \alpha _{n}^{2}

diverge, the product ∏n=1∞(1+αn)\displaystyle \prod_{n=1}^{\infty } (1+\alpha_{n}) converges, where

αn={−1k,if n=2k−1,1k+1k+1kk,if n=2k. \alpha _{n}= \begin{cases} -\dfrac {1}{\sqrt{k}}, & \text{if } n=2k-1, \\ \dfrac {1}{\sqrt{k}}+\dfrac {1}{k}+\dfrac {1}{k\sqrt{k}}, & \text{if } n=2k. \end{cases}
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Problem 3100

Let

ζ(x)=∑n=1∞1nx \zeta (x)=\sum _{n=1}^{\infty } \frac{1}{n^{x}}

(the Riemann ζ\zeta function) and let pnp_{n} (n=1,2,⋯n=1,2,\cdots) be the sequence of primes. Prove:

∏n=1∞(1−1pnx)−1=ζ(x). \prod _{n=1}^{\infty } \left(1-\frac{1}{p_{n}^{x}}\right)^{-1}=\zeta (x).
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Problem 3101

Prove that the product

∏n=1∞(1−1pn)−1 \prod _{n=1}^{\infty } \left(1-\frac{1}{p_{n}}\right)^{-1}

and the series

∑n=1∞1pn \sum _{n=1}^{\infty } \frac{1}{p_{n}}

(where pnp_{n}, n=1,2,⋯n=1,2,\cdots, is the sequence of primes) diverge (Euler).

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Problem 3102

Let an>0a_{n}>0 (n=1,2,⋯n=1,2,\cdots), and

anan+1=1+pn+O(1n1+C)(C>0). \frac{a_{n}}{a_{n+1}}=1+\frac{p}{n}+O\left(\frac{1}{n^{1+C}}\right) \quad (C>0).

Prove that an=O∗(1np)a_{n}=O^{*}\left(\dfrac {1}{n^{p}}\right).

Hint: investigate lim⁡n→∞annp=a1∏n=1∞an+1an(1+1n)p\displaystyle \lim_{n \rightarrow \infty } a_{n}n^{p}=a_{1}\prod_{n=1}^{\infty } \dfrac {a_{n+1}}{a_{n}}\left(1+\dfrac {1}{n}\right)^{p}.

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Problem 3103

Use Wallis's formula to prove

1⋅3⋅5⋯(2n−1)2⋅4⋅6⋯(2n)∼1πn. \frac{1 \cdot 3 \cdot 5\cdots (2n-1)}{2 \cdot 4 \cdot 6\cdots (2n)} \sim \frac{1}{\sqrt{\pi n}}.
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Problem 3104

Prove that the expression

an=n! ennn+12 a_{n}=\frac{n! \, e^{n}}{n^{n+\frac{1}{2}}}

has a nonzero limit AA as n→∞n \rightarrow \infty.

Deduce from this Stirling's formula

n!=Ann+12e−n(1+εn), n!=An^{n+\frac{1}{2}}e^{-n}(1+\varepsilon _{n}),

where lim⁡n→∞εn=0\displaystyle \lim_{n \rightarrow \infty } \varepsilon_{n}=0 and A=2πA=\sqrt{2\pi }.

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Problem 3105

According to Euler's definition, the gamma function Γ(x)\Gamma (x) is determined by the following formula:

Γ(x)=lim⁡n→∞n! nxx(x+1)⋯(x+n). \Gamma (x)=\lim _{n \rightarrow \infty } \frac{n! \, n^{x}}{x(x+1)\cdots (x+n)}.

Starting from this formula:

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(a)

Express the function Γ(x)\Gamma (x) in the form of an infinite product;

(b)

Prove that Γ(x)\Gamma (x) is meaningful for all real xx that are not negative integers;

(c)

Deduce the following property

Γ(x+1)=xΓ(x); \Gamma (x+1)=x\Gamma (x);
(d)

Find the value of Γ(n)\Gamma (n) for positive integer nn.

Problem 3106

Let the function f(x)f(x) be integrable on the closed interval [a,b][a,b], and

δn=b−an,fin=f(a+iδn)(i=1,2,⋯ ,n), \delta _{n}=\frac{b-a}{n},\quad f_{in}=f(a+i\delta _{n}) \quad (i=1,2,\cdots ,n),

Prove

lim⁡n→∞∏i=1n(1+δnfin)=e∫abf(x) dx. \lim _{n \rightarrow \infty } \prod _{i=1}^{n} (1+\delta _{n}f_{in})=e^{\int _{a}^{b} f(x)\, dx}.
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Problem 3107

Prove that:

lim⁡n→∞∏i=0n−1(a+ib)n1n∑i=0n−1(a+ib)=2e, \lim _{n \rightarrow \infty } \frac{\sqrt[n]{\displaystyle \prod _{i=0}^{n-1} (a+ib)}}{\dfrac {1}{n}\displaystyle \sum _{i=0}^{n-1} (a+ib)}=\frac{2}{e},

where a>0a>0 and b>0b>0.

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Problem 3108

Let fn(x)f_{n}(x) (n=1,2,⋯n=1,2,\cdots) be continuous functions on the interval (a,b)(a,b), and

∣fn(x)∣≤cn(n=1,2,⋯ ), \left|f_{n}(x)\right| \leq c_{n} \quad (n=1,2,\cdots ),

where the series ∑n=1∞cn\displaystyle \sum_{n=1}^{\infty } c_{n} converges. Prove that the function

F(x)=∏n=1∞[1+fn(x)](∣fn(x)∣<1) F(x)=\prod _{n=1}^{\infty } \left[1+f_{n}(x)\right] \quad (\left|f_{n}(x)\right|<1)

is continuous on the interval (a,b)(a,b).

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Problem 3109

Find an expression for the derivative of the function

F(x)=∏n=1∞[1+fn(x)] F(x)=\prod _{n=1}^{\infty } \left[1+f_{n}(x)\right]

What are sufficient conditions for F′(x)F'(x) to exist?

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Problem 3110

Prove that if 0<x<y0<x<y, then

lim⁡n→∞x(x+1)⋯(x+n)y(y+1)⋯(y+n)=0. \lim _{n \rightarrow \infty } \frac{x(x+1)\cdots (x+n)}{y(y+1)\cdots (y+n)}=0.
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Section 5.10Stirling's Formula[10/100%]
Problem 3111

Using Stirling's formula, approximately compute: lg⁡100!\lg 100!.

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Problem 3112

Using Stirling's formula, approximately compute: 1⋅3⋅5⋯19991 \cdot 3 \cdot 5\cdots 1999.

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Problem 3113

Using Stirling's formula, approximately compute: 1⋅3⋅5⋯992⋅4⋅6⋯100\dfrac {1 \cdot 3 \cdot 5\cdots 99}{2 \cdot 4 \cdot 6\cdots 100}.

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Problem 3114

Using Stirling's formula, approximately compute: C10040C_{100}^{40}.

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Problem 3115

Using Stirling's formula, approximately compute: 100!20!⋅30!⋅50!\dfrac {100!}{20! \cdot 30! \cdot 50!}.

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Problem 3116

Using Stirling's formula, approximately compute: ∫01(1−x2)50 dx\displaystyle \int_{0}^{1} (1-x^{2})^{50}\, dx.

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Problem 3117

Using Stirling's formula, approximately compute: ∫02πsin⁡200x dx\displaystyle \int_{0}^{2\pi } \sin^{200}x\, dx.

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Problem 3118

For the product

(2n−1)!!=1⋅3⋅5⋯(2n−1) (2n-1)!!=1 \cdot 3 \cdot 5\cdots (2n-1)

derive an asymptotic formula.

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Problem 3119

If nn is very large, approximately compute C2nnC_{2n}^{n}.

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Problem 3120

Using Stirling's formula find the following limits:

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(a)

lim⁡n→∞n!n2\displaystyle \lim_{n \rightarrow \infty } \sqrt[n^{2}]{n!};

(b)

lim⁡n→∞nn!n\displaystyle \lim_{n \rightarrow \infty } \frac{n}{\sqrt[n]{n!}};

(c)

lim⁡n→∞n(2n−1)!!n\displaystyle \lim_{n \rightarrow \infty } \frac{n}{\sqrt[n]{(2n-1)!!}};

(d)

lim⁡n→∞ln⁡n!ln⁡nn\displaystyle \lim_{n \rightarrow \infty } \frac{\ln n!}{\ln n^{n}}.

Section 5.11Approximating Continuous Functions by Polynomials[16/100%]
Problem 3121

Using the given data, construct the polynomial Pn(x)P_{n}(x) of lowest degree nn:

xx−2-2004455
yy5511−3-311

What are Pn(−1)P_{n}(-1), Pn(1)P_{n}(1), Pn(6)P_{n}(6) approximately equal to?

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Problem 3122

Using the Lagrange interpolation formula, write the equation of the parabola

y=ax2+bx+c y=ax^{2}+bx+c

passing through the three points A(x0−h,y−1)A(x_{0}-h,y_{-1}), B(x0,y0)B(x_{0},y_{0}), C(x0+h,y1)C(x_{0}+h,y_{1}).

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Problem 3123

Using the values x0=1x_{0}=1, y0=1y_{0}=1; x1=25x_{1}=25, y1=5y_{1}=5; x2=100x_{2}=100, y2=10y_{2}=10, derive an approximate formula for the square root y=xy=\sqrt{x} (1≤x≤1001 \leq x \leq 100).

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Problem 3124

Using the values sin⁡0∘=0\sin 0^{\circ }=0, sin⁡30∘=12\sin 30^{\circ }=\dfrac {1}{2}, sin⁡90∘=1\sin 90^{\circ }=1, derive the approximate formula

sin⁡x∘≈ax+bx3(0≤x≤90). \sin x^{\circ } \approx ax+bx^{3} \quad (0 \leq x \leq 90).

Using this formula, approximately find: sin⁡20∘\sin 20^{\circ }, sin⁡40∘\sin 40^{\circ }, sin⁡80∘\sin 80^{\circ }.

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Problem 3125

Taking the values xi=0,±12,±1x_{i}=0,\pm \dfrac {1}{2},\pm 1 as the interpolation points of a Lagrange polynomial, construct the Lagrange interpolation polynomial on the closed interval [−1,1]\left[-1,1\right] for the function f(x)=∣x∣f(x)=\left|x\right|.

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Problem 3126

Using a Lagrange polynomial in place of the function y(x)y(x), approximately compute

∫02y(x) dx, \int _{0}^{2} y(x) \, dx,

where

xx000.50.5111.51.522
y(x)y(x)554.54.5332.52.555
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Problem 3127

For the functions xx, x2x^{2}, x3x^{3}, construct on the closed interval [0,1]\left[0,1\right] the Bernstein polynomial

Bn(x)=∑i=0nf(in)Cnixi(1−x)n−i. B_{n}(x)=\sum _{i=0}^{n} f\left(\frac{i}{n}\right)C_{n}^{i}x^{i}(1-x)^{n-i}.
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Problem 3128

For a given function f(x)f(x) on the closed interval [a,b]\left[a,b\right], write out the formula for the Bernstein polynomial Bn(x)B_{n}(x).

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Problem 3129

On the interval [−1,1]\left[-1,1\right], approximate the function f(x)=∣x∣+x2f(x)=\dfrac {\left|x\right|+x}{2} using the Bernstein polynomial B4(x)B_{4}(x). Plot the graphs of y=∣x∣+x2y=\dfrac {\left|x\right|+x}{2} and y=B4(x)y=B_{4}(x).

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Problem 3130

On −1≤x≤1-1 \leq x \leq 1, approximate the function using an even-degree Bernstein polynomial

f(x)=∣x∣. f(x)=\left|x\right|.
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Problem 3131

For the function f(x)=ekxf(x)=e^{kx} (a≤x≤ba \leq x \leq b), write out the Bernstein polynomial Bn(x)B_{n}(x).

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Problem 3132

On the closed interval −π2≤x≤π2-\dfrac {\pi }{2} \leq x \leq \dfrac {\pi }{2}, for the function f(x)=cos⁡xf(x)=\cos x, compute the Bernstein polynomial Bn(x)B_{n}(x).

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Problem 3133

Prove: on the closed interval [−1,1]\left[-1,1\right], ∣x∣=lim⁡n→∞Pn(x)\left|x\right|=\displaystyle \lim_{n \rightarrow \infty } P_{n}(x), where

Pn(x)=1−1−x22−∑i=2n1⋅3⋯(2i−3)2⋅4⋯(2i)(1−x2)i. P_{n}(x)=1-\frac{1-x^{2}}{2}-\sum _{i=2}^{n} \frac{1 \cdot 3\cdots (2i-3)}{2 \cdot 4\cdots (2i)}(1-x^{2})^{i}.
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Problem 3133.1

Let f(x)∈C[a,b]f(x) \in C\left[a,b\right], Mk=∫abxkf(x) dx=0M_{k}=\displaystyle \int_{a}^{b} x^{k}f(x)\, dx=0 (k=0,1,2,⋯k=0,1,2,\cdots). Prove: for x∈[a,b]x \in \left[a,b\right], f(x)≡0f(x) \equiv 0.

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Problem 3133.1

Let f(x)∈C[a,b]f(x) \in C\left[a,b\right], Mk=∫abxkf(x) dx=0M_{k}=\displaystyle \int_{a}^{b} x^{k}f(x)\, dx=0 (k=0,1,2,⋯k=0,1,2,\cdots). Prove: for x∈[a,b]x \in \left[a,b\right], f(x)≡0f(x) \equiv 0.

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Problem 3134

Let f(x)f(x) be a continuous function on −π≤x≤π-\pi \leq x \leq \pi, and let ana_{n}, bnb_{n} (n=0,1,2,⋯n=0,1,2,\cdots) be its Fourier coefficients. Prove that the Fejer trigonometric polynomial

σn(x)=a02+∑i=1n−1(1−in)(aicos⁡ix+bisin⁡ix) \sigma _{n}(x)=\frac{a_{0}}{2}+\sum _{i=1}^{n-1} \left(1-\frac{i}{n}\right)(a_{i}\cos ix+b_{i}\sin ix)

converges uniformly to the function f(x)f(x) on the interval (−π,π)(-\pi ,\pi ).

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Problem 3135

Construct the Fejer polynomial σ2n−1(x)\sigma_{2n-1}(x) for the following function:

f(x)=∣x∣(−π≤x≤π). f(x)=\left|x\right| \quad (-\pi \leq x \leq \pi ).
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