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Summing an infinite series

  1. Apr 18, 2016 #1
    1. The problem statement, all variables and given/known data

    Show that ##\sum_{n=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}##.

    2. Relevant equations

    3. The attempt at a solution

    ##\frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots##.

    Do I now factorise?
  2. jcsd
  3. Apr 18, 2016 #2
    No, I'm pretty sure there's no way to directly perform the summation in this form.

    You can either make use of the integral form of the Riemann zeta function or a neat trick using Fourier series (Parseval's theorem).
  4. Apr 18, 2016 #3

    Ray Vickson

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    Your "equation"
    [tex] \frac{1}{n^{4}} = \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots [/tex]
    is wrong. The only time it could be correct is if ##n = 1## and you include only one term on the right-hand-side.

    The solution to your problem cannot involve just pre-calculus methods, but instead, very likey involves advanced methods in calculus that use matrrial beyond that found in first or second courses in calculus.
  5. Apr 18, 2016 #4
    A typo!
  6. Apr 18, 2016 #5

    Ray Vickson

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    OK, but the rest of my answer applies unchanged.
    Last edited by a moderator: Apr 19, 2016
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