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:


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

Consider the point $x = \pi$.

By Fourier's Theorem:

Thus: