prove-that-n-1-1-n-1-n-2-pi-2-12-

Question Number 170003 by mathlove last updated on 13/May/22

p r o v e t h a t

\underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{n^{2}} = \frac{π^{2}}{12}

Commented by mathlove last updated on 14/May/22

a n y o n e s o l v e t h i s

Answered by ajfour last updated on 14/May/22

\frac{\ln (1 + x)}{x} = 1 - \frac{x}{2} + \frac{x^{2}}{3} - \frac{x^{3}}{4} + . .

\Rightarrow

\int_{0}^{^{1}} \frac{\ln (1 + x) d x}{x} = \frac{1}{1^{2}} - \frac{1}{2^{2}} + \frac{1}{3^{2}} - \frac{1}{4^{2}} + . .

= I = \int_{0}^{^{1}} \frac{\ln (1 + x) d x}{x} = \frac{π^{2}}{12}

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