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Fourier series problem

  1. May 17, 2005 #1
    Hi, I need help on the following problem on Fourier series:

    Let phi(x)=1 for 0<x<pi. Expand
    [tex] 1 = \sum\limits_{n = 0}^\infty B_n cos[(n+ \frac{1}{2})x] [/tex]
    a) Find B_n.
    b) Let -2pi < x < 2pi. For which such x does this series converge? For each such x, what is the sum of the series?
    c) Apply Parseval's equality to this series. Use it to calculate the sum
    1 + 1/(3^2) + 1/(5^2) + 1/(7^2) + ....

    I know the formula for B_n for a function's Fourier series on the interval -L < x < L, so in this question I need to do some kind of odd or even extension for parts a and b, but I don't know how. Please help. Thanks.
  2. jcsd
  3. May 17, 2005 #2
    Ok, actually for part a, I'm pretty sure I should do an even extension of the function at x=0 so that it runs from -pi to pi. And I can then determine the B_n's.

    But for part b, it looks like I need to somehow extend to -2pi to 2pi. I looked in books, and does an odd extension at x=pi so that function is -1 from pi to 2i. But I don't understand why. And how would I find the x's for the series to converge?
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