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,