Sum of Reciprocals of Squares Alternating in Sign/Proof 3

Proof
Let $f \left({x}\right)$ be the real function defined on $\left({-\pi \,.\,.\, \pi}\right)$ as:


 * $f \left({x}\right) = \pi^2 - x^2$

By Fourier Series: $\pi^2 - x^2$ over $\left({-\pi \,.\,.\, \pi}\right)$:


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

for $x \in \left({-\pi \,.\,.\, \pi}\right)$.

Setting $x = 0$: