1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Verify that a sum converges to particular function (Fourier Series)
Loading...