Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign

Proof
By Fourier Series for $x \left({\pi - x}\right)$ over $\left[{0 \,.\,.\, \pi}\right]$:


 * $\displaystyle x \left({\pi - x}\right) = \frac 8 \pi \sum_{r \mathop = 0}^\infty \frac {\sin \left({2 n + 1}\right) x} {\left({2 n + 1}\right)^3}$

for $x \in \left[{0 \,.\,.\, \pi}\right]$.

Setting $x = \dfrac \pi 2$:

whence the result.