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Evaluate infinite sum using Parseval's theorem (Fourier series)

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that: [itex]\sum_{n=1}^{\infty}\frac{1}{n^4} = \frac{π^4}{90}[/itex]
    Hint: Use Parseval's theorem

    2. Relevant equations
    Parseval's theorem:

    [itex]\frac{1}{\pi}\int_{-\pi}^{\pi} |f(x)|^2dx = \frac{a_0^2}{2}+\sum_{n=1}^{\infty}(a_n^2+b_n^2)[/itex]

    3. The attempt at a solution
    I've been trying to solve this for ages and I just can't figure out what to do. I know you're supposed to use Parseval's theorem. All I've managed to do was plug in [itex]\frac{1}{n^4}[/itex] into the summation part of the Parseval's equation and I substituted the formula for a0 but I couldn't get very far.

    Any help would be really appreciated.
  2. jcsd
  3. Nov 12, 2011 #2
    Try the function [itex]f(x)=x^2[/itex] for [itex]x\in [-\pi,\pi][/itex].
  4. Nov 12, 2011 #3
    It works, ty
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