I need to compute Fourier series for the following function: $f(x)=\frac{-\pi}{4} $ for $-\pi \leq x <0$, and $\frac{\pi}{4} $ for $ 0 \leq x \leq \pi$, and then to use it and compute $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$
I tried to use Parseval equality:
$\widehat{f(n)}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}=\frac{1}{4in}-\frac{(-1)^n}{4in}, \sum_{-\infty}^{\infty}|\widehat{f(n)}|^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^2.$
$\sum_{-\infty}^{\infty}|\frac{1}{4in}-\frac{(-1)^n}{4in}|=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)=\frac{\pi^2}{16}.$
Does someone see how can I compute form that the requsted sum?
Thanks!