I was given to expand in a Fourier series the function $f(x)=|x|, \; x \in [-\pi, \pi]$. The Fourier series is quite known and I had done the calculations and I ended up to the formula:
$$f(x)=\frac{\pi}{2}+2 \sum_{n=1}^{\infty}\frac{(-1)^n -1}{\pi n^2}\cos n x$$
which is correct. Now the exercise wants me to evaluate the series: $$ \sum_{n={\rm odd} \geq 1}^{\infty}\frac{1}{n^2}$$
What value should I insert in the Fourier? Plugging $x=0$ I get another series , that is I get $\zeta(2)$. For the series I want , the value i should plug in is still is a mystery to me.
If you have: $$ |x| = \frac{\pi}{2}-\frac{4}{\pi}\sum_{n\geq 0}\frac{\cos((2n+1)x)}{(2n+1)^2}\tag{1}$$ by setting $x=0$ it follows that: $$ \sum_{n\geq 0}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}.\tag{2}$$