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Math Help - Is the Theorem Correct?

  1. #1
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    Is the Theorem Correct?

    Nevermind, I just forgot what cosine is at pi/2...
    Last edited by ragnar; August 8th 2010 at 08:43 PM.
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    ... Well, even though I solved that last part, I'm now stuck trying to find the value of \displaystyle \sum^{\infty}_{n=1} \frac{1}{(2n-1)^{2}}. I kind of suspect I should be able to get this from Parseval's Theorem, given in Apostol as \frac{1}{\pi} \displaystyle \int^{2 \pi}_{0} |f(x)|^{2} dx = \frac{a_{0}^{2}}{2} + \sum^{\infty}_{n=1} (a_{n}^{2} + b_{n}^{2}). I suspect this because, somehow, the teacher claimed to have used this to show that \displaystyle \sum^{\infty}_{n=1} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}.

    Anyway, I plugged in the constant function \frac{\pi}{2} and got a truism, so in general I can tell I don't want to use constant functions. I suppose I want to find some function such that a_{n}^{2} + b_{n}^{2} is equal to the sum I'm looking to evaluate, but I can't see how to make this magically happen.
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Notice that

    \frac{1}{2^2}\left(\frac{1}{1^2}+\frac{1}{2^2}+\fr  ac{1}{3^2}+\dots\right) + \left(\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\d  ots\right)

     = \frac{1}{1^2}+\frac{1}{2^2}+\dots =\frac{\pi^2}{6},

    from which you can just solve. (Note that all manipulations are justified by absolute convergence of the series.)
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    I'm afraid I don't follow. If this is a proof that \displaystyle \sum^{\infty}_{n=1} \frac{1}{n^{2}} = \frac{\pi^{2}}{6} I'm not exactly sure what is taken as known, from which we prove this. However, here is my (somewhat failed) attempt at reproducing the professor's argument: Using Parseval, \frac{\pi^{2}}{3} = \frac{a_{0}}{2} + \displaystyle \sum^{\infty}_{n=1}(a_{n}^{2} + b_{n}^{2}) where a_{0} = 2\pi and a_{n} = 0 and b_{n} = \frac{2}{n} so \frac{\pi^{2}}{3} = 2\pi + \displaystyle 4\sum^{\infty}_{n=1} \frac{1}{n^{2}} \Rightarrow I get the wrong answer.

    In any case, I've used another proof of the same fact, and I'm still not sure how to solve the problem I'm faced with.
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