I'm trying to use the Fourier series for $f(x)=x^3$ on $[-\pi,\pi]$ to show that
$$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2k-1)^3}=\frac{\pi^3}{32} $$
I've found the Fourier series to be
$$S(f)(x)= 2\sum_{k=1}^{\infty} \frac{(-1)^k6+(-1)^{k+1}\pi^2k^2}{k^3} \sin(kx)$$
I've tried evaluating at $x=\frac{\pi}{2}$ to get the correct result, but the my equation does not simplify correctly. The even $k$ terms vanish because of the $\sin(kx)$, but I can't seem to get rid of the "$\pi^2$" on the righthand side.
Suggestions or hints would be nice. Thanks!
Best Answer
Use the Fourier series of $x^3 - \pi^2 x$ instead.