Fourier series technique to show that the following series sum to the quantities

[maddogtheman]

Use the Fourier series technique to show that the following series sum to the quantities shown:
1+1/3^2+1/5^2+...+1/n^2=pi^2/8 for n going to infinity

I foudn the series to be:

sum(1/(2n-1)^2,n,1,infinity)

but I don't know how to prove the idenity.

I don't know how to go about solving it using the Fourier method. Any help would be greatly appreciated, thanks!

I was able to prove sum(1/n^4,n,1,infinity)=pi^4/90 and sum(1/n^2,n,1,infinity)=pi^2/6 and I'm not sure if the problem is simular.

 

[fresh_42]

Set $f(x)=|x|$. Then $a_0=\int_{-\pi}^{\pi} |x|\,dx =\pi^2$ and $a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}|x|\cos(nx)\,dx =\frac{\cos(\pi n)-1}{n^2}$ and consider $x=0$.