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Fourier Series

  1. Mar 19, 2008 #1
    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.
     
  2. jcsd
  3. Mar 23, 2008 #2
    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.
     
  4. Mar 27, 2008 #3
    Thanks

    Thanks I got it
     
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