Using Parseval's Identity

Question

Using parsevals Identity we have obtained that

$$t = \sum_{n=-\infty}^\infty \frac{i(-1)^ne^{-int}}{n} $$,

and $c_0=0$

, prove that $\frac{\pi^2}{6}= \sum\frac{1}{n^2}$.

I am really struglling this problem. I have integate both sides to get $t^2/2 \mid^?_? = \sum \frac{(-1)^ne^{-int}}{n^2}\mid^?_?$, however any evauation point does not appear to work.

Answer 1

Parseval's identity tells us that

$$\frac1{2\pi}\int_{-\pi}^\pi |f(t)|^2dt=\sum_{n=-\infty}^\infty |c_n|^2$$

Do this and you get exactly what you need (Basel Problem solved)