What function do I pick for $\sum_{n=1}^{\infty}(1/n^6)$?

Question

Possible Duplicate:
Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

What function do I pick for the summation from $$\sum_{n =1}^{\infty}\frac{1}{n^6} \ ?$$ using Parseval's identity

Answer 1

Hint:

1. In Parseval's formula you will square the Fourier coefficients $c_n$.

2. Can you prove for a generic function $f$ that $$c_n(f)= j_n+\frac{t}{n}c_n(df/dx)$$ for suitable numbers $j_n$ and $t$.

2'. Can you prove for a generic function $f$ that $$c_n(f)= j_n+\frac{k_n}{n}+\frac{t}{n^2}c_n(d^2f/dx^2)$$ for suitable numbers $j_n,\,k_n$ and $t$.

2''. Can you prove for a generic function $f$ that $$c_n(f)= j_n+\frac{k_n}{n}+\frac{l_n}{n^2}+\frac{t}{n^3}c_n(d^3f/dx^3)$$ for suitable numbers $j_n,\,k_n,\,l_n$ and $t$.

I hope you see the picture.