Summing an infinite series

  • #1
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Homework Statement



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

Homework Equations



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?
 

Answers and Replies

  • #2
954
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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).
 
  • #3
Ray Vickson
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Homework Statement



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

Homework Equations



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?
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.
 
  • #4
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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.
A typo!
 
  • #5
Ray Vickson
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A typo!
OK, but the rest of my answer applies unchanged.
 
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