Compute infinite sum using fourier series

Question

I'm trying to tackle the following question:

Use Fourier series of the function $f(x)=x(\pi+|x|)$ in $[-\pi,\pi]$ to compute the infinite sum $$ \sum_{n=1}^{\infty}\frac{(-1)^n}{(2n-1)^3}$$

So, I found that the coefficients of the series are $\displaystyle c_n=\begin{cases}0,& n\text{ even} \\ \frac{8}{in^3},& n\text{ odd}\end{cases}$.

How should I continue? how do the coefficients help me compute the sum?

Please help, thank you!

Answer 1

That is a special case of a general identity proved here.

With a Fourier-analytic approach, we may notice that: $$ \sum_{n\geq 1}\frac{(-1)^{n+1} \sin((2n-1)x)}{(2n-1)^2} $$ is the Fourier sine series of a triangle wave, hence: $$ \sum_{n\geq 1}\frac{(-1)^n}{(2n-1)^3} = -\int_{0}^{\pi/2}\left(\frac{\pi x}{4}\right)\,dx = \color{red}{-\frac{\pi^3}{32}}.$$

Answer 2

You can use Poisson summation formula on $F$ which extends $f$ to be zero outside $[-\pi,\pi]$.