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Math Help - Deriving Sum from Fourier Series

  1. #1
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    Deriving Sum from Fourier Series

    I've worked out the following Fourier series for exp(x) (valid between -pi and pi) and would now like to be able to derive the sum of 1/(1-n^2).

    e^x = \frac{1}{\pi} sinh(\pi) + \sum_{1}^{\infty} \frac{2sinh(\pi)}{\pi(1+n^2)} (-1)^n(cos(nx)-nsin(nx))

    This would work perfectly if I could substitute pi or -pi into the series, but unfortunately it is not valid for these values so gives the wrong result. Subbing in 0 works but this gives the alternating series (-1)^n/(1-n^2).

    Can anyone give me a push in the right direction? I've tried differentiating/integrating the series but this does not seem to be helpful.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by StaryNight View Post
    I've worked out the following Fourier series for exp(x) (valid between -pi and pi) and would now like to be able to derive the sum of 1/(1-n^2).

    e^x = \frac{1}{\pi} sinh(\pi) + \sum_{1}^{\infty} \frac{2sinh(\pi)}{\pi(1+n^2)} (-1)^n(cos(nx)-nsin(nx))

    This would work perfectly if I could substitute pi or -pi into the series, but unfortunately it is not valid for these values so gives the wrong result. Subbing in 0 works but this gives the alternating series (-1)^n/(1-n^2).

    Can anyone give me a push in the right direction? I've tried differentiating/integrating the series but this does not seem to be helpful.

    Thanks in advance.
    You need to Use Dirchlet conditions at the end points

    Dirichlet conditions - Wikipedia, the free encyclopedia

    The Fourier series converges to the average of the right and left limit at that point so for your series you get
    \displaystyle f(\pi)=\frac{\lim_{x \to \pi^-}e^{x}+\lim_{x \to \pi^+}e^{x}}{2}=\frac{e^{\pi}+e^{-\pi}}{2}=\cosh(\pi)
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    You need to Use Dirchlet conditions at the end points

    Dirichlet conditions - Wikipedia, the free encyclopedia

    The Fourier series converges to the average of the right and left limit at that point so for your series you get
    \displaystyle f(\pi)=\frac{\lim_{x \to \pi^-}e^{x}+\lim_{x \to \pi^+}e^{x}}{2}=\frac{e^{\pi}+e^{-\pi}}{2}=\cosh(\pi)
    Many thanks, I was not aware of this theorem.
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