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Math Help - Fourier Series problem

  1. #1
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    Fourier Series problem

    find a half-range series of  f(x)=x^{2}  \ \ in \ \ (0,1)

    and then show that:

     \sum_{n=1}^{\infty}{\frac{1}{n^{2}}}=\frac{\pi^{2}  }{6}

    i got a series:

     x^{2}=\frac{4}{3}+\sum_{n=1}^{\infty}{\frac{4(-1)^{n}}{n^{2}\pi^{2}}\cos(n\pi x) }
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  2. #2
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    Quote Originally Posted by silversand View Post
    find a half-range series of  f(x)=x^{2} \ \ in \ \ (0,1)

    and then show that:

     \sum_{n=1}^{\infty}{\frac{1}{n^{2}}}=\frac{\pi^{2}  }{6}

    i got a series:

     x^{2}=\frac{4}{3}+\sum_{n=1}^{\infty}{\frac{4(-1)^{n}}{n^{2}\pi^{2}}\cos(n\pi x) }
    How did you get the lead term \frac{4}{3} ?
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  3. #3
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    Quote Originally Posted by silversand View Post
    find a half-range series of  f(x)=x^{2}  \ \ in \ \ (0,1)

    and then show that:

     \sum_{n=1}^{\infty}{\frac{1}{n^{2}}}=\frac{\pi^{2}  }{6}

    i got a series:

     x^{2}=\frac{4}{3}+\sum_{n=1}^{\infty}{\frac{4(-1)^{n}}{n^{2}\pi^{2}}\cos(n\pi x) }
    your answer seems to have the wrong constant term
    you should get 1/3 (a0 is 2/3 and you want half of it)

    see the following page for explanation (take l = 1 in your problem)
    half range fourier series of f(x) = x in (0,l)

    for the deduction
    put x = pi

    for the formulae use http://keral2008.blogspot.com/2009/0...er-series.html
    Last edited by qpmathelp; May 28th 2009 at 08:20 AM.
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