Section 2.13

Properties of Convergent Series

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Subsection 2.13.2Problems[20/50%]
Problem 2.13.1

Find the nn-th partial sum SnS_{n} of the series and the sum SS of this series.

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(1)

25+225+…+25n+…\frac{2}{5}+\frac{2}{25}+\ldots +\frac{2}{5^{n}}+\ldots;

(2)

1−13+19−127+…+(−1)n−13n−1+…1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\ldots +\frac{(-1)^{n-1}}{3^{n-1}}+\ldots;

(3)

(13+15)+(132+152)+…+(13n+15n)+…\left(\frac{1}{3}+\frac{1}{5}\right)+\left(\frac{1}{3^{2}}+\frac{1}{5^{2}}\right)+\ldots +\left(\frac{1}{3^{n}}+\frac{1}{5^{n}}\right)+\ldots;

(4)

(3+12)+(32−16)+(34+118)+…+(32n−1+(−1)n−12⋅3n−1)+…\left(3+\frac{1}{2}\right)+\left(\frac{3}{2}-\frac{1}{6}\right)+\left(\frac{3}{4}+\frac{1}{18}\right)+\ldots +\left(\frac{3}{2^{n-1}}+\frac{(-1)^{n-1}}{2 \cdot 3^{n-1}}\right)+\ldots;

(5)

(110+2102+5103)+(1102+2103+5104)+…+(110n+210n+1+510n+2)+…\left(\frac{1}{10}+\frac{2}{10^{2}}+\frac{5}{10^{3}}\right)+\left(\frac{1}{10^{2}}+\frac{2}{10^{3}}+\frac{5}{10^{4}}\right)+\ldots +\left(\frac{1}{10^{n}}+\frac{2}{10^{n+1}}+\frac{5}{10^{n+2}}\right)+\ldots

Problem 2.13.2

Find the nn-th partial sum SnS_{n} of the series and the sum SS of this series.

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(1)

13⋅4+14⋅5+…+1(n+2)(n+3)+…\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\ldots +\frac{1}{(n+2)(n+3)}+\ldots;

(2)

11⋅4+14⋅7+…+1(3n−2)(3n+1)+…\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\ldots +\frac{1}{(3 n-2)(3 n+1)}+\ldots;

(3)

11⋅2⋅3+12⋅3⋅4+…+1n(n+1)(n+2)+…\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\ldots +\frac{1}{n(n+1)(n+2)}+\ldots;

(4)

13⋅5⋅7+15⋅7⋅9+…+1(2n+1)(2n+3)(2n+5)+…\frac{1}{3 \cdot 5 \cdot 7}+\frac{1}{5 \cdot 7 \cdot 9}+\ldots +\frac{1}{(2 n+1)(2 n+3)(2 n+5)}+\ldots;

(5)

1a(a+1)(a+2)(a+3)+1(a+1)(a+2)(a+3)(a+4)+…\frac{1}{a(a+1)(a+2)(a+3)}+\frac{1}{(a+1)(a+2)(a+3)(a+4)}+\ldots

…+1(a+n)(a+n+1)(a+n+2)(a+n+3)+… \ldots +\frac{1}{(a+n)(a+n+1)(a+n+2)(a+n+3)}+\ldots
Problem 2.13.3

Find the nn-th partial sum SnS_{n} of the series and the sum SS of this series.

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(1)

∑n=1∞116n2−8n−3\sum_{n = 1}^{\infty } \frac{1}{16 n^{2}-8 n-3};

(2)

∑n=1∞125n2+5n−6\sum_{n = 1}^{\infty } \frac{1}{25 n^{2}+5 n-6};

(3)

∑n=1∞136n2−24n−5\sum_{n = 1}^{\infty } \frac{1}{36 n^{2}-24 n-5};

(4)

∑n=1∞149n2+7n−12\sum_{n = 1}^{\infty } \frac{1}{49 n^{2}+7 n-12}

Problem 2.13.4

Find the nn-th partial sum SnS_{n} of the series and the sum SS of this series.

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(1)

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

(2)

∑n=1∞14n2+4n−3\sum_{n = 1}^{\infty } \frac{1}{4 n^{2}+4 n-3};

(3)

∑n=1∞116n2−8n−15\sum_{n = 1}^{\infty } \frac{1}{16 n^{2}-8 n-15};

(4)

∑n=1∞136n2+12n−35\sum_{n = 1}^{\infty } \frac{1}{36 n^{2}+12 n-35}.

Problem 2.13.5

Find the nn-th partial sum SnS_{n} of the series and the sum SS of this series.

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(1)

∑n=1∞3n−1n(n+1)(n+2)(n+3)(n+4)\sum_{n = 1}^{\infty } \frac{3 n-1}{n(n+1)(n+2)(n+3)(n+4)};

(2)

∑n=1∞2n+9n(n+1)(n+2)(n+3)(n+4)(n+5)\sum_{n = 1}^{\infty } \frac{2 n+9}{n(n+1)(n+2)(n+3)(n+4)(n+5)};

(3)

∑n=1∞2n+1n2(n+1)2\sum_{n = 1}^{\infty } \frac{2 n+1}{n^{2}(n+1)^{2}}

(4)

∑n=1∞n(2n−1)2(2n+1)2\sum_{n = 1}^{\infty } \frac{n}{(2 n-1)^{2}(2 n+1)^{2}};

(5)

∑n=1∞n−n2−1n(n+1)\sum_{n = 1}^{\infty } \frac{n-\sqrt{n^{2}-1}}{\sqrt{n(n+1)}};

(6)

∑n=1∞(n+2−2n+1+n)\sum_{n = 1}^{\infty }(\sqrt{n+2}-2 \sqrt{n+1}+\sqrt{n}).

Problem 2.13.6

Find the nn-th partial sum SnS_{n} of the series and the sum SS of this series.

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(1)

∑n=2∞ln⁡(1−1n2)\sum_{n = 2}^{\infty } \ln \left(1-\frac{1}{n^{2}}\right);

(2)

∑n=2∞ln⁡(1−2n(n+1))\sum_{n = 2}^{\infty } \ln \left(1-\frac{2}{n(n+1)}\right);

(3)

∑n=2∞ln⁡n3−1n3+1\sum_{n = 2}^{\infty } \ln \frac{n^{3}-1}{n^{3}+1};

(4)

∑n=1∞ln⁡n(2n+1)(n+1)(2n−1)\sum_{n = 1}^{\infty } \ln \frac{n(2 n+1)}{(n+1)(2 n-1)}

(5)

∑n=1∞sin⁡12ncos⁡32n\sum_{n = 1}^{\infty } \sin \frac{1}{2^{n}} \cos \frac{3}{2^{n}}

(6)

∑n=1∞sin⁡α2n+1cos⁡3α2n+1\sum_{n = 1}^{\infty } \sin \frac{\alpha }{2^{n+1}} \cos \frac{3 \alpha }{2^{n+1}};

(7)

∑n=0∞1n!(n+2)\sum_{n = 0}^{\infty } \frac{1}{n !(n+2)};

(8)

∑n=1∞arctg⁡12n2\sum_{n = 1}^{\infty } \operatorname {arctg} \frac{1}{2 n^{2}}.

Problem 2.13.7

Prove that if ak=1ukuk+1…uk+ma_{k} = \frac{1}{u_{k} u_{k+1} \ldots u_{k+m}}, where u1,u2,…,un,…u_{1}, u_{2}, \ldots , u_{n}, \ldots are consecutive terms of an arithmetic progression with common difference dd, with d≠0,uk≠0(k∈N)d \neq 0, u_{k} \neq 0(k \in N), then the series ∑k=1∞ak\sum_{k = 1}^{\infty } a_{k} converges and

∑n=1∞1unun+1…un+m=1mdu1u2…um \sum _{n = 1}^{\infty } \frac{1}{u_{n} u_{n+1} \ldots u_{n+m}} = \frac{1}{m d u_{1} u_{2} \ldots u_{m}}
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Problem 2.13.8

Find the sum of the series.

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(1)

∑n=1∞1n(n+2)(n+3)\sum_{n = 1}^{\infty } \frac{1}{n(n+2)(n+3)};

(2)

∑n=1∞2−nn(n+1)(n+3)\sum_{n = 1}^{\infty } \frac{2-n}{n(n+1)(n+3)};

(3)

∑n=1∞1(n+1)(n+2)(2n+1)(2n+5)\sum_{n = 1}^{\infty } \frac{1}{(n+1)(n+2)(2 n+1)(2 n+5)};

(4)

∑n=1∞3n+4n(n+1)(n+4)\sum_{n = 1}^{\infty } \frac{3 n+4}{n(n+1)(n+4)}.

Problem 2.13.9

Find the sum of the series.

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(1)

∑n=1∞((−1)n3n+i5n)\sum_{n = 1}^{\infty }\left(\frac{(-1)^{n}}{3^{n}}+\frac{i}{5^{n}}\right);

(2)

∑n=1∞(1+i)n4n\sum_{n = 1}^{\infty } \frac{(1+i)^{n}}{4^{n}};

(3)

∑n=1∞n+2+inn(n+1)(n+2)\sum_{n = 1}^{\infty } \frac{n+2+i n}{n(n+1)(n+2)};

(4)

∑n=1∞n(1−i)n\sum_{n = 1}^{\infty } \frac{n}{(1-i)^{n}}.

Problem 2.13.10

Find the sum of the series.

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(1)

∑n=1∞ancos⁡nα,α∈R,a∈R,∣a∣<1\sum_{n = 1}^{\infty } a^{n} \cos n \alpha , \alpha \in R, a \in R,\left|a\right| < 1;

(2)

∑n=1∞ansin⁡nα,α∈R,a∈R,∣a∣<1\sum_{n = 1}^{\infty } a^{n} \sin n \alpha , \alpha \in R, a \in R,\left|a\right| < 1.

Problem 2.13.11

Prove the divergence of the series using the necessary condition for convergence.

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(1)

∑n=1∞(−1)n2n2+3n+42n2+1\sum_{n = 1}^{\infty }(-1)^{n} \frac{2 n^{2}+3 n+4}{2 n^{2}+1};

(2)

∑n=1∞3n+45n+1\sum_{n = 1}^{\infty } \sqrt{\frac{3 n+4}{5 n+1}};

(3)

∑n=1∞n+2n3+2n+43\sum_{n = 1}^{\infty } \frac{n+2}{\sqrt[3]{n^{3}+2 n+4}}

(4)

∑n=1∞0,02n\sum_{n = 1}^{\infty } \sqrt[n]{0,02}

(5)

∑n=1∞(n−1n+1)n\sum_{n = 1}^{\infty }\left(\frac{n-1}{n+1}\right)^{n}

(6)

∑n=1∞(3n3−23n3+4)n3\sum_{n = 1}^{\infty }\left(\frac{3 n^{3}-2}{3 n^{3}+4}\right)^{n^{3}}.

Problem 2.13.12

Prove the divergence of the series using the necessary condition for convergence.

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(1)

∑n=1∞n3ln⁡2(n+1)\sum_{n = 1}^{\infty } \frac{\sqrt[3]{n}}{\ln^{2}(n+1)};

(2)

∑n=1∞(n2+2)ln⁡n2+1n2\sum_{n = 1}^{\infty }\left(n^{2}+2\right) \ln \frac{n^{2}+1}{n^{2}}

(3)

∑n=1∞(n+1)arctg⁡1n+2\sum_{n = 1}^{\infty }(n+1) \operatorname {arctg} \frac{1}{n+2};

(4)

∑n=1∞n3+1n+3arcsin⁡1n2+2\sum_{n = 1}^{\infty } \frac{n^{3}+1}{n+3} \arcsin \frac{1}{n^{2}+2};

(5)

∑n=1∞1ln⁡(n+1)n\sum_{n = 1}^{\infty } \frac{1}{\sqrt[n]{\ln (n+1)}}

(6)

∑n=1∞nn+1/n(n+1/n)n\sum_{n = 1}^{\infty } \frac{n^{n+1 / n}}{(n+1 / n)^{n}}.

Problem 2.13.13

Using the Cauchy criterion, prove the convergence of the series ∑n=1∞an\sum_{n = 1}^{\infty } a_{n}, if:

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(1)

an=cos⁡nx2na_{n} = \frac{\cos n x}{2^{n}}

(2)

an=sin⁡nαn(n+1)a_{n} = \frac{\sin n \alpha }{n(n+1)}

(3)

an=cos⁡αnn2a_{n} = \frac{\cos \alpha^{n}}{n^{2}}

(4)

an=cos⁡nx−cos⁡(n+1)xna_{n} = \frac{\cos n x-\cos (n+1) x}{n}.

Problem 2.13.14

Using the Cauchy criterion, prove the divergence of the series ∑n=1∞an\sum_{n = 1}^{\infty } a_{n}, if:

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(1)

an=12n+1a_{n} = \frac{1}{2 n+1};

(2)

an=n+1n2+4a_{n} = \frac{n+1}{n^{2}+4};

(3)

an=1n(n+1)a_{n} = \frac{1}{\sqrt{n(n+1)}};

(4)

an=ln⁡(1+1/n)a_{n} = \ln (1+1 / n).

Problem 2.13.15

Prove that if the series ∑n=1∞zn\sum_{n = 1}^{\infty } z_{n} and ∑n=1∞wn\sum_{n = 1}^{\infty } w_{n} converge and their sums equal ss and σ\sigma, respectively, then the series ∑n=1∞ζn\sum_{n = 1}^{\infty } \zeta_{n}, where ζn=λzn+μwn\zeta_{n} = \lambda z_{n}+\mu w_{n}, also converges for any complex λ,μ\lambda , \mu, and its sum equals λs+μσ\lambda s+\mu \sigma.

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Problem 2.13.16

Prove that:

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(1)

if the series ∑n=1∞zn\sum_{n = 1}^{\infty } z_{n} converges, then for any mm its mm-th remainder converges;

(2)

if any remainder of the series converges, then the series itself converges.

In this case the equality S=Sm+rmS = S_{m}+r_{m} holds, where SS and SmS_{m} are, respectively, the sum and the mm-th partial sum of the original series, and rmr_{m} is the sum of its mm-th remainder.

Problem 2.13.17

Prove that if the series ∑n=1∞zn\sum_{n = 1}^{\infty } z_{n} converges and its sum equals SS, then the series ∑j=1∞wj\sum_{j = 1}^{\infty } w_{j} also converges, where w1=z1+z2+…+zk1,w2=zk1+1+zk1+2+…+zk2,…w_{1} = z_{1}+z_{2}+\ldots +z_{k_{1}}, \quad w_{2} = z_{k_{1}+1}+z_{k_{1}+2}+\ldots +z_{k_{2}}, \ldots

…,wj=zkj−1+1+zkj−1+2+…+zkj \ldots , \quad w_{j} = z_{k_{j-1}+1}+z_{k_{j-1}+2}+\ldots +z_{k_{j}}

({kj}\left\{ k_{j}\right\} is a strictly increasing sequence of natural numbers) and moreover ∑j=1∞wj=S\sum_{j = 1}^{\infty } w_{j} = S.

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Problem 2.13.18

Prove that if the series ∑n=1∞an2\sum_{n = 1}^{\infty } a_{n}^{2}, where an∈R(n∈N)a_{n} \in R(n \in N), converges, then the series ∑n=1∞∣an∣n\sum_{n = 1}^{\infty } \frac{\left|a_{n}\right|}{n} also converges.

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Problem 2.13.19

Determine what can be said about the convergence of the series ∑n=1∞cn\sum_{n = 1}^{\infty } c_{n}, where cn=an+bn(n∈N)c_{n} = a_{n}+b_{n}(n \in N), if it is known that:

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(1)

the series ∑n=1∞an\sum_{n = 1}^{\infty } a_{n} converges, while the series ∑n=1∞bn\sum_{n = 1}^{\infty } b_{n} diverges;

(2)

both of these series diverge.

Problem 2.13.20

Prove that if the series ∑n=1∞an2\sum_{n = 1}^{\infty } a_{n}^{2} and ∑n=1∞bn2\sum_{n = 1}^{\infty } b_{n}^{2}, where an∈R,bn∈R(n∈N)a_{n} \in R, b_{n} \in R (n \in N), converge, then the following series also converge:

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(1)

∑n=1∞∣anbn∣\sum_{n = 1}^{\infty }\left|a_{n} b_{n}\right|

(2)

∑n=1∞(an+bn)2\sum_{n = 1}^{\infty }\left(a_{n}+b_{n}\right)^{2}.