Sum of Reciprocals of Squares of Odd Integers/Proof 6

Proof
Let $f \left({x}\right)$ be the real function defined on $\left({0 \,.\,.\, 2 \pi}\right)$ as:


 * $f \left({x}\right) = \begin{cases}

-\pi & : 0 < x \le \pi \\ x - \pi & : \pi < x < 2 \pi \end{cases}$

By Fourier Series: $-\pi$ over $\left({0 \,.\,.\, \pi}\right)$, $x - \pi$ over $\left({\pi \,.\,.\, 2 \pi}\right)$, its Fourier series can be expressed as:


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

Consider the point $x = \pi$.

By Dirichlet's Theorem for 1-Dimensional Fourier Series:

Thus: