Proving the Sum of a Series using Fourier Series Technique

[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.

 

[coomast]

Calculate the Fourier series of the following function:
$$f(x)=|x| \qquad -\pi<x<\pi$$
meaning:
$$f(x)=-x \qquad -\pi<x<0$$
$$f(x)=x \qquad 0<x<\pi$$
with period $2\pi$. After this set x=0 in the resulting series and you arrive at the result.

 

[maddogtheman]

Thanks

Thanks I got it