find-n-1-1-n-n-2-

Question Number 38108 by maxmathsup by imad last updated on 21/Jun/18

f i n d \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n^{2}}

Commented by prof Abdo imad last updated on 24/Jun/18

l e t S = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n^{2}}

S = \sum_{n = 1}^{\infty} \frac{1}{4 n^{2}} - \sum_{n = 0}^{\infty} \frac{1}{{(2 n + 1)}^{2}}

= \frac{1}{4} \frac{π^{2}}{6} - \frac{π^{2}}{8} = \frac{π^{2}}{24} - \frac{3 π^{2}}{24} = - \frac{π^{2}}{12}

S = - \frac{π^{2}}{12} .