Answer to Question #109842 in Real Analysis for Chinmoy Kumar Bera

Question

the following series 1-1/2^2-1/3^2+1/4^2-1/5^2-1/6^2+... is convergent or not?

Answer 1

$1-\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^2}-\frac{1}{5^2}-\frac{1}{6^2}+...(*)$

Consider a series of modules

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...=\\ =\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$

A series $\sum\limits_{n=1}^{\infty}\frac{1}{n^{\alpha}}$ is convergent if $\alpha>1$ ,

 divergent if $\alpha \leq1$ .

 In our case $\alpha=2>1$ ,

 so the series of modules is convergent, hence the series (*) is

 also convergent.