Find the fourier series of the following function: $$f:[-\pi,\pi]\to \mathbb{R} , f(x) = \pi x - x^3$$ Easy task. $a_n = 0$ since f is odd and $b_n = \dfrac{2\pi}{n}(-1)^{n+1} + \dfrac{2{\pi}^2}{n}(-1)^{n+2} +\dfrac{12}{n^3}(-1)^{n+3}$
Hopefully no mistakes have I made there.
So, $f(x) \approx \sum_{n=0}^{\infty} b_n \sin nx$
Prove the following results: $$a) \sum_{n=0}^{\infty} \dfrac{(-1)^n}{(2n+1)^3} = \dfrac{{\pi}^3}{32} $$ $$b)\sum_{n=1}^{\infty} \dfrac{1}{n^6} = \dfrac{{\pi}^6}{945}$$
I tried Parseval's identity, but it doesn't seem to lead to anything remarkable. Any tips?