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Find SUM for fourier series

  1. Oct 1, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]
    h(x)=\left\{\begin{matrix}
    9+2x , 0<x<\pi\\ -9+2x , -pi<x<0

    \end{matrix}\right.
    \\
    Find \ the \ sum \ of \ the \ fourier \ series \ for \ x=\frac{3\pi}{2} and\ x=\pi
    \\
    The \ fourier \ series \ is:
    \\
    h(x)=9+\pi + \sum_{n=1}^{inf} \frac{18-2(9+2\pi)(-1)^2}{n\pi}sin(nx)
    [/itex]
    Also period is 2pi.

    3. The attempt at a solution
    I have already calculated the fourier series. I however don't know how I can find the sum for the x's. I have searched the internet and looked in my book but I can't find any examples that help me. I'm stuck and don't know how to go from here.
     
  2. jcsd
  3. Oct 1, 2012 #2

    jbunniii

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    Can you simplify [itex]\sin(n \frac{3\pi}{2})[/itex] and [itex]\sin(n \pi)[/itex]?
     
  4. Oct 1, 2012 #3
    [itex]\sin(n \frac{3\pi}{2})[/itex] changes for different n's. Example: -1, 0 and 1. So I don't know how to simplify that. However [itex]\sin(n \pi)[/itex] will always be 0 for all n.
     
  5. Oct 1, 2012 #4

    jbunniii

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    [itex]\sin(n \frac{3\pi}{2})[/itex] is zero for even [itex]n[/itex]. Try a change of variables so that you will only sum over the odd values of [itex]n[/itex].
     
  6. Oct 1, 2012 #5
    I don't quite understand how to do that. I tried the sum function at wolframalpha and I get that it doesnt converge.
     
  7. Oct 1, 2012 #6

    LCKurtz

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    You don't work this kind of problem by actually summing the series. You use the convergence theorem. Doesn't your text have a theorem something to the effect that for a periodic function f(x) satisfying the Dirichlet conditions the series converges to f(x) at points where f is continuous and to ##\frac {f(x^+) + f(x^-)} 2## at points ##x## where there is a jump discontinuity? So you can answer the question by examining the periodic extension of ##f(x)##. In fact, you don't even need to calculate the Fourier Series to answer the question.
     
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