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Verify that a sum converges to particular function (Fourier Series)

  1. Sep 21, 2009 #1
    1. The problem statement, all variables and given/known data
    Verify the formula x=2*(sin(x)-(1/2)sin(2x)+(1/3)sin(3x)-...), {x,-Pi,Pi}

    2. Relevant equations

    3. The attempt at a solution
    I guess, I am to show that the sum on the right converges to the function x. I began by rewriting the sum on the RHS as [tex]$\displaystyle\sum_{k=1}^k 2*\frac{1}{k}*sin(kx)(-1)^{k-1}$[/tex]

    Now, I'm not sure what I am to do next. Am I to take the limit of k to infinity? If so how does one solve that? Thank you!

    Also, if one graphs this in Mathematica, it can be seen that as k becomes larger and larger the sin function becomes more and more like x through the origin between -Pi and Pi. I believe, however, that I am to show this algebraicly...
    Last edited: Sep 21, 2009
  2. jcsd
  3. Sep 21, 2009 #2


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    Do you know how to calculate Fourier series coefficients? I assume all you're supposed to do is verify that they match what you're given
  4. Sep 21, 2009 #3
    Ah, yes I do! I was assuming that since the sum was written that was supposed to use that. I guess if I attempt to calculate the function with only sine terms the coefficients would be given by: [tex]b_{k}[/tex]=[tex]\frac{2}{Pi}[/tex][tex]\int^{Pi}_{0}f(x)sin(kx)dx[/tex]. You obtain (-1)^(k-1)(2/k) for the coefficient. This then gives you the Fourier Series that gives you the RHS of the original equation. But I am now to prove convergence? That is something I am having difficulty understanding. Any help would be appreciated.
    Last edited: Sep 21, 2009
  5. Sep 22, 2009 #4
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