Math Problems Vault

Problem 1585

By direct division in ascending powers one can establish the identity:

\frac{1}{1-x} = 1+x+x^{2}+\ldots +x^{n-1}+\frac{x^{n}}{1-x}.

Starting from it, show that the series

1+x+x^{2}+\ldots

converges for $\left|x\right| < 1$ and its sum equals $\dfrac {1}{1-x}$ .

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Problem 1586

Using the identity $\dfrac {1}{\nu }-\dfrac {1}{\nu +1} = \dfrac {1}{\nu (\nu +1)}$ , prove that the series, whose convergence is to be established directly,

\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots

converges and has sum equal to unity.

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Problem 1587

Prove the equality

\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots = \frac{1}{2}.

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Problem 1588

Prove the equality

\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}+\ldots = \frac{1}{4}.

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Problem 1589

Using the hint of problem 1586 ( $\dfrac {1}{\nu }-\dfrac {1}{\nu +1} = \dfrac {1}{\nu (\nu +1)}$ ), find the sum

\frac{1}{1 \cdot 4}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 6}+\frac{1}{4 \cdot 7}+\frac{1}{5 \cdot 8}+\ldots

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Problem 1590

Prove the equality

\sum _{n=1}^{\infty }(v_{n}-v_{n+1}) = v_{1}-v,

where it is assumed that $\lim_{n \to \infty } v_{n} = v$ .

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Problem 1591

Having established the equality

\frac{m}{(m+1)(m+2)}+\frac{m}{(m+2)(m+3)}+\frac{m}{(m+3)(m+4)}+\ldots = \frac{m}{m+1},

prove that in this case, as $m \to \infty$ , the limit of the sum of the series is not equal to the sum of the limits of the terms of the series, namely: $\lim_{m \to \infty }(u_{1}+u_{2}+u_{3}+\ldots ) = 1$ , while

\lim u_{1}+\lim u_{2}+\lim u_{3}+\ldots = 0+0+0+\ldots = 0.

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Problem 1592

Investigate the convergence of the series for various values of the variable $x$ :

x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{4}+\ldots

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Problem 1593

Investigate the convergence of the series for various values of the variable $x$ :

x+x^{4}+x^{16}+x^{25}+\ldots

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Problem 1594

Investigate the convergence of the series for various values of the variable $x$ :

1-\frac{x}{3}+\frac{x^{2}}{5}-\frac{x^{3}}{7}+\ldots

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Problem 1595

Investigate the convergence of the series for various values of the variable $x$ :

\frac{\sin x}{1 \cdot 2}+\frac{\sin 2x}{2 \cdot 3}+\frac{\sin 3x}{3 \cdot 4}+\ldots

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Problem 1596

Investigate the convergence of the series for various values of the variable $x$ :

x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\ldots

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Problem 1597

Investigate the convergence of the series for various values of the variable $x$ :

\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+\frac{1}{(x+2)(x+3)}+\ldots

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Problem 1598

Investigate the convergence of the series for various values of the variable $x$ :

\frac{1}{x^{2}}+\frac{1}{x^{2}+1^{2}}+\frac{1}{x^{2}+2^{2}}+\ldots

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Problem 1599

Investigate the convergence of the series:

1+x+\frac{x^{2}}{1 \cdot 2}+\frac{x^{3}}{1 \cdot 2 \cdot 3}+\ldots

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Problem 1600

Investigate the convergence of the series:

1+x+1 \cdot 2x^{2}+1 \cdot 2 \cdot 3x^{3}+\ldots

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Problem 1601

Investigate the convergence of the series:

x+2x^{2}+x^{3}+4x^{4}+5x^{5}+\ldots

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Problem 1602

Investigate the convergence of the series:

x+\frac{2x^{2}}{3}+\frac{2^{2}x^{3}}{5}+\frac{2^{3}x^{4}}{7}+\ldots

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Problem 1603

Investigate the convergence of the series:

x+\frac{1}{2^{2}}x^{2}+\frac{1 \cdot 2}{3^{3}}x^{3}+\frac{1 \cdot 2 \cdot 3}{4^{4}}x^{4}+\ldots

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Problem 1604

Investigate the convergence of the series:

x+\frac{1 \cdot 2}{1 \cdot 3}x^{2}+\frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5}x^{3}+\frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7}x^{4}+\ldots

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Problem 1605

Investigate the convergence of the series:

\frac{x-1}{2}+\frac{1}{\sqrt{2}}\left(\frac{x-1}{2}\right)^{2}+\frac{1}{\sqrt{3}}\left(\frac{x-1}{2}\right)^{3}+\ldots

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Problem 1606

Investigate the convergence of the series:

1+\frac{1}{2x\sqrt{2}}+\frac{1}{4x^{2}\sqrt{3}}+\frac{1}{8x^{3}\sqrt{4}}+\frac{1}{16x^{4}\sqrt{5}}+\ldots

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Problem 1607

Investigate the convergence of the series for $a > 0$ , $b > 0$ :

\frac{x-3}{a}+\frac{(x-3)^{2}}{a+b}+\frac{(x-3)^{3}}{a+2b}+\frac{(x-3)^{4}}{a+3b}+\ldots

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Problem 1608

Investigate the convergence of the series:

\frac{x}{x+2}-\frac{1}{3}\left(\frac{x}{x+2}\right)^{3}+\frac{1}{5}\left(\frac{x}{x+2}\right)^{5}-\ldots

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Problem 1609

Investigate the convergence of the series:

\frac{1+x}{x+\frac{1}{x}}+\frac{5+x}{2\left(x+\frac{1}{x}\right)^{2}}+\frac{25+x}{4\left(x+\frac{1}{x}\right)^{3}}+\frac{125+x}{8\left(x+\frac{1}{x}\right)^{4}}+\ldots

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Problem 1610

Investigate the convergence of the series:

\tg x-\frac{\tg ^{2}x}{5}+\frac{\tg ^{3}x}{9}-\frac{\tg ^{4}x}{13}+\ldots

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Problem 1611

Investigate the convergence of the series:

1+\frac{2\sin x}{3}+\frac{4\sin ^{2}x}{5}+\frac{8\sin ^{3}x}{7}+\frac{16\sin ^{4}x}{9}+\ldots

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Problem 1612

Investigate the convergence of the series:

x+\frac{x^{4}}{1 \cdot 2}+\frac{x^{9}}{1 \cdot 2 \cdot 3}+\ldots = \sum _{n=1}^{\infty }\frac{x^{n^{2}}}{n!}.

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Problem 1613

Investigate the convergence of the series:

\frac{1}{e^{x}-1}+\frac{2}{e^{2x}-1}+\frac{3}{e^{3x}-1}+\ldots

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Problem 1614

Investigate the convergence of the series $\sum_{n=1}^{\infty } u_{n}$ , if $u_{1}=1$ , and $\dfrac {u_{n+1}}{u_{n}} = \dfrac {3}{4}+\dfrac {(-1)^{n}}{2}$ .

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Problem 1615

The series $u_{1}+u_{2}+u_{3}+\ldots$ is formed by the following rule: $u_{n} = \dfrac {1}{2^{m\sigma }}$ , where $2^{m} \leq n < 2^{m+1}$ , and $\sigma$ is a constant number. Prove that for $\sigma > 1$ the series $u_{1}+u_{2}+u_{3}+\ldots$ converges, and its sum equals $\dfrac {2^{\sigma }}{2^{\sigma }-2}$ .

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Problem 1616

Using the preceding result (problem 1615), prove that for $\sigma > 1$ the series $1+\dfrac {1}{2^{\sigma }}+\dfrac {1}{3^{\sigma }}+\dfrac {1}{4^{\sigma }}+\ldots$ converges.

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Problem 1617

The terms of a series are formed by the rule:

u_{n} = \frac{1}{2^{m\sigma }},

where $2^{m-1} < n \leq 2^{m}$ , and $\sigma$ is a constant number. Prove that for $0 < \sigma < 1$ the series $u_{1}+u_{2}+u_{3}+\ldots$ diverges.

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Problem 1618

Prove that the series $1+\dfrac {1}{2^{\sigma }}+\dfrac {1}{3^{\sigma }}+\dfrac {1}{4^{\sigma }}+\ldots$ diverges for $0 < \sigma \leq 1$ .

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Problem 1619

Establish that the series $1+\dfrac {1}{\sqrt{2}}+\dfrac {1}{\sqrt{3}}+\dfrac {1}{\sqrt{4}}+\ldots$ is divergent, relying on the inequality

\frac{1}{\sqrt{n}} > \frac{2}{\sqrt{n}+\sqrt{n+1}} = 2\left(\sqrt{n+1}-\sqrt{n}\right).

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Problem 1620

Investigate the convergence of the series:

1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\ldots

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Problem 1621

Investigate the convergence of the series:

1+\frac{1}{2} \cdot \frac{1}{2}+\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{3}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{1}{4}+\ldots

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Problem 1622

Investigate the convergence of the series:

1+\frac{2}{1} \cdot \frac{1}{2}+\frac{2 \cdot 4}{1 \cdot 3} \cdot \frac{1}{3}+\frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5} \cdot \frac{1}{4}+\ldots

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Problem 1623

Investigate the convergence of the series for $m > 0$ :

1-\frac{m}{1}+\frac{m(m-1)}{1 \cdot 2}-\frac{m(m-1)(m-2)}{1 \cdot 2 \cdot 3}+\ldots

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Problem 1624

(a)(b)

Investigate the convergence of the series:

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(a)

$1-\dfrac {1}{3}+\dfrac {1}{5}-\dfrac {1}{6}+\ldots$ ;

(b)

$1-\dfrac {1}{2^{\sigma }}+\dfrac {1}{3^{\sigma }}-\dfrac {1}{4^{\sigma }}+\ldots$ ; $\sigma > 0$ .

Problem 1625

Investigate the convergence of the series for $m > -1$ :

1+\frac{m}{1}+\frac{m(m-1)}{1 \cdot 2}+\frac{m(m-1)(m-2)}{1 \cdot 2 \cdot 3}+\ldots

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Problem 1626

Investigate the convergence of the series:

1-\frac{2}{1} \cdot \frac{1}{2}+\frac{2 \cdot 4}{1 \cdot 3} \cdot \frac{1}{3}-\frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5} \cdot \frac{1}{4}+\ldots

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Problem 1627

Investigate the convergence of the series:

\sin x-\sin \sin x+\sin \sin \sin x-\ldots

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Problem 1628

Investigate the convergence of the series:

\frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+\frac{\sin 4x}{4}+\ldots

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Problem 1629

Investigate the convergence of the series:

\cos x+\frac{\cos 3x}{3}+\frac{\cos 5x}{5}+\frac{\cos 7x}{7}+\ldots

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Problem 1630

Investigate the convergence of the series:

1+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\frac{1}{7}+\frac{1}{8}-\frac{2}{9}+\ldots

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Problem 1631

Investigate the convergence of the series:

1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}+\ldots

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Problem 1632

Using the fundamental criterion of convergence, prove that the sum of an absolutely convergent series does not change if the same terms of the series are arranged in any other order.

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Problem 1633

Show that the sum of a non-absolutely convergent series does not change if the terms of the series are rearranged so that none of them is moved from its place by more than $m$ places, where $m$ is any given number.

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Problem 1634

Prove the equality

1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\ldots = 2\left(1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-\frac{1}{10}-\frac{1}{12}+\ldots \right),

thereby showing that for non-absolutely convergent series the sum depends essentially on the order of the terms.

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Problem 1635

Using the equalities

s = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\ldots = 1-\left(\frac{1}{2}-\frac{1}{3}\right)-\left(\frac{1}{4}-\frac{1}{5}\right)-\ldots ,

s_{1} = 1-\frac{1}{2}+\frac{1}{3}+\frac{1}{5}-\frac{1}{4}+\frac{1}{7}+\frac{1}{9}-\frac{1}{6}+\ldots = 1-\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{5}-\frac{1}{4}\right)+\left(\frac{1}{7}+\frac{1}{9}-\frac{1}{6}\right)+\ldots ,

prove that $s < 1$ , while $s_{1} > 1$ , and hence $s_{1} \neq s$ .

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Problem 1636

Prove that the series $1-\dfrac {1}{\sqrt{2}}+\dfrac {1}{\sqrt{3}}-\dfrac {1}{\sqrt{4}}+\dfrac {1}{\sqrt{5}}-\dfrac {1}{\sqrt{6}}+\ldots$ is convergent, while the series $1+\dfrac {1}{\sqrt{3}}-\dfrac {1}{\sqrt{2}}+\dfrac {1}{\sqrt{5}}+\dfrac {1}{\sqrt{7}}-\dfrac {1}{\sqrt{4}}+\dfrac {1}{\sqrt{9}}+\dfrac {1}{\sqrt{11}}-\dfrac {1}{\sqrt{6}}+\ldots$ , obtained from the first by rearranging the terms, is divergent.

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Problem 1637

Prove that the series

x(1-x)+x^{2}(1-x)+x^{3}(1-x)+\ldots

converges on the segment $0 \leq x \leq 1$ , but not uniformly.

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Problem 1638

Prove that the series

1+x+\frac{x^{2}}{1 \cdot 2}+\frac{x^{3}}{1 \cdot 2 \cdot 3}+\ldots

converges for every $x$ , but not uniformly. The same series converges uniformly on any finite segment $a \leq x \leq b$ .

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Problem 1639

Prove that the series

\sum _{n=1}^{\infty } \frac{1}{x^{2}+n^{2}}

converges uniformly on the whole real axis $-\infty < x < \infty$ .

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Problem 1640

Prove that the series

\sin x+\frac{\sin 2x}{2^{3}}+\frac{\sin 3x}{3^{3}}+\ldots

can be differentiated term by term.

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Problem 1641

Prove that the series

\frac{\cos x}{1 \cdot 2}+\frac{\cos 2x}{2 \cdot 3}+\frac{\cos 3x}{3 \cdot 4}+\ldots

represents a continuous function of $x$ .

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Problem 1642

Using Abel's identity, prove that the series

a_{1}\sin x+a_{2}\sin 2x+a_{3}\sin 3x+\ldots ,

a_{1}\cos x+a_{2}\cos 2x+a_{3}\cos 3x+\ldots

when the conditions

a_{1} \geq a_{2} \geq a_{3} \geq a_{4} \geq \ldots , \quad \lim _{n \to \infty } a_{n} = 0

hold, converge uniformly on every segment containing neither inside it nor at its endpoints the values $x = 0, \pm 2\pi , \pm 4\pi , \ldots$ .

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Problem 1643

Prove that the sum of the series

s(x) = \sum _{n=1}^{\infty } \frac{\sin 2^{n^{2}}x}{a^{n^{2}}}, \quad a > 1,

represents a continuous function.

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Problem 1644

Prove that for $a > 2$ the sum of the series of problem 1643, $s(x) = \sum_{n=1}^{\infty } \dfrac {\sin 2^{n^{2}}x}{a^{n^{2}}}$ , has derivative:

s'(x) = \sum _{n=1}^{\infty } \left(\frac{2}{a}\right)^{n^{2}}\cos 2^{n^{2}}x.

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Problem 1645

Prove that for $1 < a < 2$ the function $s(x)$ of problem 1644 has no derivative (while nevertheless being continuous).

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