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Summation proof

  1. Aug 13, 2009 #1

    Mentallic

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    [tex]\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}[/tex]

    I'd like to know how to prove this summation. And if possible, what is the significance of having [itex]\pi[/itex] in the answer?
     
  2. jcsd
  3. Aug 13, 2009 #2

    uart

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    If you're familar with Fourier series then one nice method is to consider the expansion of a "saw-tooth" wave as follows.

    Let [itex]y(x) = \pi x[/itex] : [itex]-0.5 \leq x \leq 0.5[/itex]

    Now make it periodic as per [itex]y(x) = y(x-k)[/itex] : [itex]-0.5+k \leq x \leq 0.5+k[/itex], for all integer k.

    It's fairly easy to show that the Fourier series expansion is,

    [tex]y = \sin(2 \pi x) - \frac{1}{2} \sin(4 \pi x) \, ... \, + \frac{(-1)^{k+1}}{k} \sin(2k \pi x) + \, ...[/tex]

    Consider the mean squared value of y, calculated two different ways. Firstly calculate directly from y(x),

    [tex]MS(y) = \int_{x=-0.5}^{+0.5} (\pi x)^2 dx = \frac {\pi^2}{12}[/tex]

    Now repeating the calculation but this time using the Fourier series (and making use of the fact that the terms are orthagonal) we get,

    [tex]MS(y) =0.5 ( 1 + 1/4 + 1/9 + ... 1/k^2 + ... )[/tex]

    Equating these two expressions for the mean squared value gives the required sum.
     
    Last edited: Aug 13, 2009
  4. Aug 13, 2009 #3
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