Finding a particular value of the zeta function.

Consider the $2\pi$ -periodic function $f$ , defined by : $f(x)=\frac{\pi}{8}(\pi-|x|)$ x on $(-\pi,\pi]$ . It is easy to check that the Fourier series gives : $f(x) = \sum_{n=0}^{\infty} \frac{\sin (nx)}{(2n+1)^3}.$ Using Parseval's Identity, we get : $\sum_{n=0}^{\infty} \frac{1}{(2n+1)^6} =\frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x)^2\ \mathrm{d}x = \frac{\pi^6}{960}.$ We know that : $\sum_{n=0}^{\infty}\frac{1}{(2n+1)^6}=\sum_{n=1}^{\infty}\frac{1}{n^6}-\sum_{n=1}^{\infty}\frac{1}{(2n)^6}=\zeta(6)-\frac{\zeta(6)}{2^6}=\frac{63}{64}\zeta(6).$ Which means that : $\zeta(6)= \frac{64}{63} \cdot \frac{\pi^6}{960}= \boxed{\frac{\pi^6}{945}}.$ Any calculator should give us : $1.01734$ .

Evaluating, we get $\frac{\pi^6}{945} \approx \boxed{1.01734}$ .