Sum of Reciprocals of Squares Alternating in Sign/Proof 3

Proof
Let $\map f x$ be the real function defined on $\openint {-\pi} \pi$ as:


 * $\map f x = \pi^2 - x^2$

By Fourier Series: $\pi^2 - x^2$ over $\openint {-\pi} \pi$:


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

for $x \in \openint {-\pi} \pi$.

Setting $x = 0$: