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Math Help - Computing Sums given Fourier Series

  1. #1
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    Computing Sums given Fourier Series

    Hello,

    I am trying to compute the sum:

    Summation (1/ (n^2)): boundaries n=1 to infinity

    Given that the Fourier series for f(x) = x/2 is
    f(x) = summation {[((-1)^(n+1)) / n)]*sinnx

    I would really appreciate some help on how to perform these calculations. The text I am using doesn't give examples on these types of problems and I really don't know where to start.

    Thanks!!
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  2. #2
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    Quote Originally Posted by Chitownmegs View Post
    Hello,

    I am trying to compute the sum:

    Summation (1/ (n^2)): boundaries n=1 to infinity

    Given that the Fourier series for f(x) = x/2 is
    f(x) = summation {[((-1)^(n+1)) / n)]*sinnx
    Let f( x ) = \frac{x}{2} for -\pi< x < \pi and f(\pi)=f(-\pi)=0.
    Now define \bar f to be the periodic extension of f.

    Then (as you are given),
    \frac{x}{2} = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\sin (nx) for  0 < x < \pi

    To solve your problem try doing the following: integrate both sides and adjust the correct constant. Then substitute into this new equation x=\pi/2 and see if anything happens.
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  3. #3
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    Response to Hacker

    Thanks, Hacker. However, when integrating and substituting, I end up with pi^2/8, when I know the answer is pi^2/ 6...!

    What am I doing wrong here?
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Chitownmegs View Post
    Thanks, Hacker. However, when integrating and substituting, I end up with pi^2/8, when I know the answer is pi^2/ 6...!

    What am I doing wrong here?
    Must you do it by this series? It is much easier to use the Fourier Series for x^2 on [-1,1]. And as for why it doesn't work, did you remember to include the constant of integration?
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