Riemann Zeta Function of 4/Proof 2

Proof
By Fourier Series of x squared, for $x \in \left[{- \pi \,.\,.\, \pi}\right]$:
 * $\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \left({\left({-1}\right)^n \frac 4 {n^2} \cos n x}\right)$

Hence: