Investigation of the Convergence of Series

If $\left|u_{n}\right| < \frac{c}{n^{a}}$, where $c$ is a constant, then the series $\sum_{n=1}^{\infty } u_{n}$ converges for $a > 1$.

If $u_{n} > \frac{c}{n^{a}}$, where $c > 0$ is a constant, $a \leqslant 1$, then the series $\sum_{n=1}^{\infty } u_{n}$ diverges.

If $\left|u_{n}\right| < \frac{c}{n(\ln n)^{a}}$, where $a > 1$, then the series $\sum_{n=1}^{\infty } u_{n}$ converges, and if $\left|u_{n}\right| > \frac{c}{n(\ln n)^{a}}$, where $a \leqslant 1$, then the series $\sum_{n=1}^{\infty } u_{n}$ diverges.

Denote $\ln \ln n = \ln_{2} n$, $\ln \ln \ln n = \ln_{3} n$, , then the series $\sum u_{n}$ converges if

and diverges if

Ermakov's test. Let $f(x)$ be a positive and decreasing function for $x > x_{0}$. If for the same $x$ we have the inequality

where $\lambda < 1$, then the series $\sum_{n=1}^{\infty } u_{n}$ converges, where $u_{n} = f(n)$. If, under the same conditions, $\frac{e^{x} f(e^{x})}{f(x)} \geqslant 1$, then the series $\sum_{n=1}^{\infty } u_{n}$ diverges. Prove this statement.

Kummer's theorem. If for a given series of positive numbers $\sum u_{n}$ one can find a sequence of positive numbers $c_{n}$ such that for all $n$ we have $c_{n} u_{n} - c_{n+1} u_{n+1} > \rho u_{n+1}$, where $\rho > 0$ is a constant, then the series $\sum u_{n}$ converges. Prove this theorem.

Kummer's convergence test. If the terms of the series $\sum u_{n}$ are positive and one can find a sequence of positive numbers $c_{1}, c_{2}, c_{3}, \ldots$ such that for all $n > n_{0}$ the inequality

holds, where $a > 0$, then the given series converges.

If, on the other hand, for all $n > n_{0}$ the inequality

holds, the numbers $c_{1}, c_{2}, c_{3}, \ldots$ are positive, and the series $\sum \frac{1}{c_{n}}$ diverges, then the series $\sum u_{n}$ also diverges. Prove this statement.

Raabe's test. If $u_{n} > 0$ and $\lim_{n \rightarrow \infty } n\left(\frac{u_{n}}{u_{n+1}} - 1\right) > 1$, then the series converges. If the same limit is $< 1$, then the series diverges. Prove this statement.

Bertrand's test for the series $\sum u_{n}$, where $u_{n} > 0$ and

If $A > 1$, the series converges, and for $A < 1$ it diverges. Prove this statement.

Gauss's test for series $\sum u_{n}$ with $u_{n} > 0$ and

where $\lambda$ and $\mu$ are constants, and $\theta_{n}$ is bounded.

If $\lambda > 1$, the series converges; if $\lambda < 1$, it diverges. When $\lambda = 1$ the series converges if $\mu > 1$, and diverges for $\mu \leqslant 1$. Prove this statement.

Investigate the convergence of the series with general term $u_{n} = \frac{1}{x^{n}+1}$, $x > 0$.

Investigate the convergence of the series with general term $u_{n} = \frac{x^{n}}{1+x^{2n}}$.

Investigate the convergence of the series with general term $u_{n} = \frac{x^{n}}{(1+x)(1+x^{2})\ldots (1+x^{n})}$.

Investigate the convergence of the series with general term $u_{n} = \frac{x^{2n}}{1+x^{2n+1}}$.

Investigate the convergence of the series with general term $u_{n} = \frac{(n!)^{2}}{(2n)!} x^{n}$.

Investigate the convergence of the series with general term $u_{n} = n! \left(\frac{x}{n}\right)^{n}$.

Investigate the convergence of the series with general term $u_{n} = \frac{n^{n}}{n!} x^{n}$.

Investigate the convergence of the series with general term $u_{n} = \frac{x^{n^{2}}}{n!}$.

Investigate the convergence of the series with general term $u_{n} = \frac{\sqrt{n^{n}}}{n!} x^{n}$.

Investigate the convergence of the series with general term $u_{n} = \frac{n!}{(x+1)(x+2)\ldots (x+n)}$.

Investigate the convergence of the series with general term $u_{n} = n \frac{a(a+1)\ldots (a+n)}{b(b+1)\ldots (b+n)}$; $a > 0$, $b > 0$.

Investigate the convergence of the series with general term $u_{n} = \frac{(a+1)(2a+1)\ldots (na+1)}{(b+1)(2b+1)\ldots (nb+1)}$; $a \geqslant 0$, $b \geqslant 0$.

Investigate the convergence of the series with general term $u_{n} = \frac{\beta (\beta +1)\ldots (\beta +n-1)}{n!} \cdot \frac{1}{n^{\alpha }}$; $\alpha > 0$, $\beta > 0$.

Investigate the convergence of the series with general term $u_{n} = (2-\sqrt{e})(2-\sqrt[3]{e})\ldots (2-\sqrt[n]{e})$.

Investigate the convergence of the series with general term $u_{n} = \ln \cos \frac{x}{n}$.

Investigate the convergence of the series with general term $u_{n} = e - \left(1+\frac{1}{n}\right)^{n}$.

Investigate the convergence of the series with general term $u_{n} = \frac{1}{\sqrt{n}}\left(1-\frac{x \ln a}{n}\right)^{n}$.

Investigate the convergence of the series with general term $u_{n} = \sqrt{n+1} - 2\sqrt{n} + \sqrt{n-1}$.

Investigate the convergence of the series with general term $u_{n} = \frac{1 \cdot 3 \cdot 5 \ldots (2n-1)}{2 \cdot 4 \cdot 6 \ldots 2n} \cdot \frac{1}{2n+1}$.

Investigate the convergence of the series with general term $u_{n} = \frac{1 \cdot 3 \cdot 5 \ldots (2n-1)}{2 \cdot 4 \cdot 6 \ldots 2n} \cdot \frac{1}{\sqrt{n}}$.

Investigate the convergence of the series with general term $u_{n} = \left(\frac{1 \cdot 3 \cdot 5 \ldots (2n-1)}{2 \cdot 4 \cdot 6 \ldots 2n}\right)^{a}$.

Investigate the convergence of the hypergeometric series:

Investigate the convergence of the series $\sum u_{n}$, if $u_{1} = 1$, and

Investigate the convergence of the series:

Investigate the convergence of the series $\sum u_{n}^{-1}$, where $u_{n}$ are the Fibonacci numbers: $u_{1} = 1$, $u_{2} = 2$, $u_{3} = 3$, $u_{4} = 5$, , $u_{n+1} = u_{n}+u_{n-1}$.

Investigate the convergence of the series $1 - \frac{1}{2^{a}} + \frac{1}{3^{a}} - \frac{1}{4^{a}} + \ldots$

Investigate the convergence of the series $1 + \frac{1}{2^{a}} - \frac{2}{3^{a}} + \frac{1}{4^{a}} + \frac{1}{5^{a}} - \frac{2}{6^{a}} + \ldots$

Investigate the convergence of the series $\sin x - \sin \sin x + \sin \sin \sin x - \ldots$

Investigate the convergence of the series $\frac{\sin x}{1} + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \ldots$

Prove that the series $\nu_{0}^{a}+\nu_{1}^{a}+\nu_{2}^{a}+\ldots$, where $\nu_{0} = x$, $\nu_{1} = \sin \nu_{0}$, $\nu_{2} = \sin \nu_{1}$, , converges for $a > 2$ and diverges for $a \leqslant 2$, if $x \neq 0$, $\sin x \neq 0$.

$\nu_{1} = \operatorname {arctg} x$, $\nu_{2} = \operatorname {arctg} \nu_{1}$, $\nu_{3} = \operatorname {arctg} \nu_{2}$, Investigate the convergence of the series $\sum \nu_{m}^{a}$ for $a > 0$.

If the terms of the series $\sum_{-\infty }^{\infty } a_{n} x^{n}$ are bounded at $x = x_{0}$ and $x = x_{1}$, $\left|x_{1}\right| > \left|x_{0}\right|$, then the series converges for $\left|x_{0}\right| < \left|x\right| < \left|x_{1}\right|$. Prove this statement.

A function $f(t)$ is called convex if for all $m > 0$, $n > 0$ we have the inequality:

Prove that the sum of the series $\sum_{-\infty }^{\infty } a_{n} x^{n}$, convergent for $0 < x_{0} < x < x_{1}$, where $a_{n} > 0$, is a convex function of $t = \ln x$.

Prove Abel's theorem: if $a_{n} > 0$ and $A_{n} = a_{1}+a_{2}+\ldots +a_{n} \rightarrow \infty$ as $n \rightarrow \infty$, then the series $\sum \frac{a_{n}}{A_{n}^{\sigma }}$ converges for $\sigma > 1$ and diverges for $\sigma \leqslant 1$.

Since the geometric mean of positive numbers is no greater than their arithmetic mean, we have

Using this, prove the inequality

where $n\alpha_{n} = a_{1}+a_{2}+\ldots +a_{n}$, and the numbers $a_{1}, a_{2}, \ldots , a_{n}$ are positive.

Under the same conditions as in problem 4999 (the numbers $a_{1}, a_{2}, \ldots , a_{m}$ are positive, $n\alpha_{n} = a_{1}+a_{2}+\ldots +a_{n}$, and the inequality $\alpha_{n}^{p} - \frac{p}{p-1}\alpha_{n}^{p-1}a_{n} \leqslant \frac{1}{p-1}\left((n-1)\alpha_{n-1}^{p} - n\alpha_{n}^{p}\right)$ has been established), show that:

If the numbers $a_{\nu }$ and $b_{\nu } > 0$, $0 < \lambda < 1$; $\lambda +\mu = 1$, then by Hölder's inequality

Using this and the result of problem 5000, prove the Hardy and Landau inequalities:

where $a_{n} > 0$, and the series on the right-hand side is convergent.

Prove Carleman's inequality:

where $u_{n} > 0$, and the series on the right-hand side converges.