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Fourier series and reimann zeta

  1. Dec 22, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the Fourier series expansion for f(x)=[itex]x^{3}[/itex], a periodic function on -[itex]\pi[/itex]<x<[itex]\pi[/itex]
    Use this to compute [itex]\zeta[/itex](6)=[itex]\sum\frac{1}{n^{6}}[/itex]


    2. Relevant equations

    Parsevals Theorom,
    Real Fourier series

    3. The attempt at a solution

    I got the Fourier series to be [itex]\sum\frac{2(-1)^{n}(6-n^{2}\pi^{2})}{n^{3}}[/itex]sin(nx)

    Using Parsevals theorom I got that [itex]\frac{\pi^{6}}{7}[/itex]=[itex]\sum\frac{4(6-(n\pi)^{2})^{2}}{n^{6}}[/itex]

    The answer is supposed to be [itex]\frac{\pi^{6}}{945}[/itex] I think, I can't see where I went wrong :S
    Thanks in advance :)
     
    Last edited: Dec 22, 2011
  2. jcsd
  3. Dec 22, 2011 #2
    Never mind, I got it.
    Just had to expand the top and use the values for the riemann zeta functions of zeta=2 and 4 and it works out :)
     
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