Tests for the convergence of series with variable sign

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.

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

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

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

Prove the convergence of the following series and find its sum: $1+\dfrac {1}{2}-\dfrac {1}{4}+\dfrac {1}{8}+\dfrac {1}{16}-\dfrac {1}{32}+\cdots .$

Prove the convergence of the following series and find its sum: $1-\dfrac {1}{2}+\dfrac {1}{3}-\dfrac {1}{4}+\dfrac {1}{5}-\dfrac {1}{6}+\cdots .$

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

Rearrange the terms of the convergent series

so that it becomes divergent.

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

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

Investigate the convergence of the variable-sign series: $1+\dfrac {1}{2}+\dfrac {1}{3}-\dfrac {1}{4}-\dfrac {1}{5}-\dfrac {1}{6}+\dfrac {1}{7}+\dfrac {1}{8}+\dfrac {1}{9}-\cdots .$

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

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

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

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

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

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

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

Prove: if

then the alternating series $b_{1}-b_{2}+b_{3}-b_{4}+\cdots +(-1)^{n-1}b_{n}+\cdots$ $(b_{n}>0)$ converges.

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

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

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

Investigate the absolute and conditional convergence of the following series: $\displaystyle \sum_{n=1}^{\infty }(-1)^{n-1} \frac{2^{n} \sin^{2n} x}{n}$.

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

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

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

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

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

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

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

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

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

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

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

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

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

Let

be a rational function, where $a_{0} \neq 0$, $b_{0} \neq 0$, and when $x \geq n_{0}$, $\left|b_{0}x^{q}+b_{1}x^{q-1}+\cdots +b_{q}\right|>0$.

Investigate the absolute and conditional convergence of the series

Investigate the convergence of the following series: $\dfrac {1}{1^{p}}-\dfrac {1}{2^{q}}+\dfrac {1}{3^{p}}-\dfrac {1}{4^{q}}+\dfrac {1}{5^{p}}-\dfrac {1}{6^{q}}+\cdots .$

Investigate the convergence of the following series: $1+\dfrac {1}{3^{p}}-\dfrac {1}{2^{p}}+\dfrac {1}{5^{p}}+\dfrac {1}{7^{p}}-\dfrac {1}{4^{p}}+\cdots .$

Investigate the convergence of the series: $1+\dfrac {1}{3^{p}}-\dfrac {1}{1^{p}}+\dfrac {1}{5^{p}}+\dfrac {1}{7^{p}}-\dfrac {1}{3^{p}}+\dfrac {1}{9^{p}}+\dfrac {1}{11^{p}}-\dfrac {1}{5^{p}}+\cdots .$

Investigate the convergence of the series: $1-\dfrac {2}{2^{q}}+\dfrac {1}{3^{p}}+\dfrac {1}{4^{p}}-\dfrac {2}{5^{q}}+\dfrac {1}{6^{p}}+\dfrac {1}{7^{p}}-\dfrac {2}{8^{q}}+\dfrac {1}{9^{p}}+\cdots .$

Prove: the series

For the series

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

For the series

determine:

Investigate the convergence of the series

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