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

  • Thread starter thaer_dude
  • Start date
  • #1
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Homework Statement


Show that: [itex]\sum_{n=1}^{\infty}\frac{1}{n^4} = \frac{π^4}{90}[/itex]
Hint: Use Parseval's theorem

Homework 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]

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.
 

Answers and Replies

  • #2
22,097
3,279
Try the function [itex]f(x)=x^2[/itex] for [itex]x\in [-\pi,\pi][/itex].
 
  • #3
19
0
It works, ty
 

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