Basel Problem/Proof 7

Proof
By Fourier Series of $x^2$, for $x \in \left[ {-\pi \,.\,.\, \pi} \right]$:


 * $\displaystyle x^2 = \frac{\pi^2} 3 + \sum_{n \mathop = 1}^\infty \left({\left( {-1} \right)^n \frac 4 {n^2} \cos n x}\right)$

Setting $x = \pi$: