Riemann Zeta Function of 4/Proof 1

Proof
By Fourier Series of Fourth Power of x, for $x \in \left[ {-\pi \,.\,.\, \pi} \right]$:
 * $\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac{8n^2\pi^2 - 48} {n^4} \cos\left( {n \pi} \right) \cos \left( {n x} \right)$

Setting $x = \pi$: