1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Summation proof

  1. Aug 13, 2009 #1


    User Avatar
    Homework Helper


    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


    User Avatar
    Science Advisor

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook