Fourier Series technique

Main Question or Discussion Point

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.
 

Answers and Replies

279
0
Calculate the Fourier series of the following function:
[tex]f(x)=|x| \qquad -\pi<x<\pi[/tex]
meaning:
[tex]f(x)=-x \qquad -\pi<x<0[/tex]
[tex]f(x)=x \qquad 0<x<\pi[/tex]
with period [itex]2\pi[/itex]. After this set x=0 in the resulting series and you arrive at the result.
 
Thanks

Thanks I got it
 

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