Riemann Zeta Function of 4/Proof 2

Proof
By Fourier Series of x squared, for $x \in \closedint {-\pi} \pi$:
 * $\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$

Hence: