I have to calculate the fourier series of $$ f(x) = \frac{(1-x/\pi)}{2} $$ On the interval $ [0,\pi] $, I got: $$ Sf(x) = \sum_{n=1}^\infty \frac{\sin{nx}}{n\pi} $$
Now, the problem begins: How to integrate (many times, maybe 3) using the Theorem for Integration of Fourier Series, and again the same calculation with the Parseval's Identity to get the known relation: $$ \sum_{n=1}^\infty \frac{1}{n⁴} = \frac{\pi^4}{90}. $$
I tried and not succeeded so far! Calculations with the Parseval's Identity gave $\frac{(2\pi)^4}{90}$, I could not get rid of this 2-factor!
Thanks!