Sum of Reciprocals of Squares of Odd Integers/Proof 6

Proof
Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as:


 * $\map f x = \begin{cases}

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

By Fourier Series: $-\pi$ over $\openint 0 \pi$, $x - \pi$ over $\openint \pi {2 \pi}$, its Fourier series can be expressed as:


 * $\map f x \sim \map S x = \displaystyle -\dfrac \pi 4 + \frac 2 \pi \sum_{r \mathop = 0}^\infty \dfrac {\cos \paren {2 r + 1} x} {\paren {2 r + 1}^2} - \sum_{n \mathop = 1}^\infty \dfrac {2 - \paren {-1}^n \sin n x} n$

Consider the point $x = \pi$.

By Fourier's Theorem:

Thus: