Q:

$ \sum_{n=0}^\infty \Bigg[ \frac{(-1)^n}{(2n+1)^3} + \frac{1}{(2n+1)^2}\Bigg] $

find the sum of the series, no solution was provided.
$$ \sum_{n=0}^\infty \Bigg[  \frac{(-1)^n}{(2n+1)^3} + \frac{1}{(2n+1)^2}\Bigg] = \sum_{n=0}^\infty   \frac{(-1)^n}{(2n+1)^3} + \sum_{n=0}^\infty   \frac{1}{(2n+1)^2}$$
I've tried a few many ways, I'll add my attempts in but they weren't correct

A:

In this mathstackexchange post, an answer shows how a Fourier expansion of $x(1-x)$ on $[0,1]$ gives $$\sum_{n=0}^\infty   \frac{(-1)^n}{(2n+1)^3}=\dfrac{\pi^3}{32}.$$ On the other hand, we have $$\sum_{n=0}^\infty   \frac{1}{(2n+1)^2}=\sum_{n=0}^\infty   \frac{1}{n^2}-\sum_{n=0}^\infty   \frac{1}{(2n)^2}=\frac{\pi^2}{6}-\frac{\pi^2}{24}=\frac{\pi^2}{8}.$$ So, the two results put together gives $$\sum_{n=0}^\infty \Bigg[  \frac{(-1)^n}{(2n+1)^3} + \frac{1}{(2n+1)^2}\Bigg] =\dfrac{\pi^3}{32}+\dfrac{\pi^2}{8}.$$

