Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign

Proof
By Half-Range Fourier Sine Series for $x \paren {\pi - x}$ over $\openint 0 \pi$:


 * $\ds x \paren {\pi - x} = \frac 8 \pi \sum_{r \mathop = 0}^\infty \frac {\sin \paren {2 n + 1} x} {\paren {2 n + 1}^3}$

for $x \in \openint 0 \pi$.

Setting $x = \dfrac \pi 2$:

whence the result.