Fourier series and reimann zeta

  • Thread starter hectorzer
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  • #1
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Homework Statement



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]


Homework Equations



Parsevals Theorom,
Real Fourier series

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:

Answers and Replies

  • #2
3
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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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