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Math Help - Show the sum

  1. #1
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    Show the sum

    Hi, can anyone help me with this question please?

    Show that:
    \sum_{n=0}^{\infty} \frac{1}{(2n+1)^6}=\frac{\pi^6}{960}

    Please help me. Thanks a lot.
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  2. #2
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    I just got a bit idea.
    Should I try to do something about partial sum? Then do something about showing the series is converging?
    But I don't know how to do it.
    Can anyone help me please?
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  3. #3
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    Quote Originally Posted by tsang View Post
    Hi, can anyone help me with this question please?

    Show that:
    \sum_{n=0}^{\infty} \frac{1}{(2n+1)^6}=\frac{\pi^6}{960}

    Please help me. Thanks a lot.
    \displaystyle 1 + \frac{1}{3^6} + \frac{1}{5^6} + .... = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \left(\frac{1}{2^6} + \frac{1}{4^6} + .... \right) = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \frac{1}{2^6} \left(1 + \frac{1}{2^6} + .... \right)

    That is,

    \displaystyle S = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \frac{S}{2^6}

    and \displaystyle 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... is a well known series .... (Google it).

    A more rigorous proof can be constructed using Fourier analysis.
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    \displaystyle 1 + \frac{1}{3^6} + \frac{1}{5^6} + .... = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \left(\frac{1}{2^6} + \frac{1}{4^6} + .... \right) = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \frac{1}{2^6} \left(1 + \frac{1}{2^6} + .... \right)

    That is,

    \displaystyle S = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \frac{S}{2^6}

    and \displaystyle 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... is a well known series .... (Google it).

    A more rigorous proof can be constructed using Fourier analysis.

    Thank you so much.
    May I ask how to use Fourier analysis please? Could you please give me some details? I'm quite bad with Fourier series, it has been one weakness for me for a long time.
    Thanks a lot.
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    \displaystyle 1 + \frac{1}{3^6} + \frac{1}{5^6} + .... = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \left(\frac{1}{2^6} + \frac{1}{4^6} + .... \right) = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \frac{1}{2^6} \left(1 + \frac{1}{2^6} + .... \right)

    That is,

    \displaystyle S = \left( 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... \right) - \frac{S}{2^6}

    and \displaystyle 1 + \frac{1}{2^6} + \frac{1}{3^6} + \frac{1}{4^6} + .... is a well known series .... (Google it).

    A more rigorous proof can be constructed using Fourier analysis.


    How can I prove in most easy way that \Sigma^\infty _{n=1} \frac{1}{n^6}=\frac{\pi^6}{945}?
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  6. #6
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    Quote Originally Posted by tsang View Post
    Thank you so much.
    May I ask how to use Fourier analysis please? Could you please give me some details? I'm quite bad with Fourier series, it has been one weakness for me for a long time.
    Thanks a lot.
    Use the Fourier series for the half-range even periodic extension of the function f(t) = -t^2 + 2t where 0 \leq t \leq 1 and apply Parseval's Theorem. (And no, I'm not going to do the calculation, that's your job if you want it).
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