Sum of Reciprocals of Squares of Odd Integers/Proof 5

Proof
By Fourier Series of Modulus of x, for $x \in \closedint {-\pi} \pi$:


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

Setting $x = \pi$: