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Fourier series and calculating a sum

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



Studying for an exam, this is a question from an earlier exam:

Calculate: sum k from 1 to infinity, 1/k^2

I have from previous question:
f(t)=pi+sum n from 1 to infinity: 2*(-1)^n/n*sin(nt)

f(t)=pi-t, -pi<=t<pi

So i need to rearrange this to calc the sum.


Homework Equations





The Attempt at a Solution



calc f(t) in a good point, I choose pi/2, use the left and right limit to get pi/2

then try top rearrange the series.


Especially I want to know mhow to do Sin(n*pi/2). it makes every second term disappear in the series but I dont know how to do this right. I tried replacing with n=2k+1 but it doesnt work.
 

Answers and Replies

  • #2
The idea behind this kind of approach is to expolit Parseval's identity.
If f(x) can be expressed in terms of a Fourier series with coefficients [itex]a_n[/itex], then you have
[tex] \int_{-\pi}^\pi |f(x)|^2 dx = \sum_{n=-\infty}^\infty |a_n|^2 [/tex]
Can you find your original series somewhere in what you have obtained through the previous question?
 

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