1. Not finding help here? Sign up for a free 30min 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!

Convergence to pi^2/6

  1. Feb 24, 2005 #1
    How does
    Sum (n^(-2)=(pi^2)/6

    Please tell me if this has been posted before (afraid :redface: )
    (in that case, i'll see the other post)
    Last edited: Feb 24, 2005
  2. jcsd
  3. Feb 24, 2005 #2
    Take f(x)=x. Then the Fourier coeffcients of f are [itex]a_n=0[/itex] and [itex]b_n=\frac{2}{n}(-1)^{n+1}[/itex]. Parseval's theorem says that:

    [tex]\frac{1}{\pi}\int_{-\pi}^\pi f^2(x)\:dx=\frac{a_0^2}{2}+\sum_{k=1}^\infty\left(a_k^2+b_k^2)[/tex]

    Since the [itex]a_n[/itex] terms are all zero, this reduces to:

    [tex]\frac{1}{\pi}\int_{-\pi}^\pi x^2\:dx=4\sum_{n=1}^\infty\frac{1}{n^2}[/tex]

    The integral is easy enough to solve, and the left hand side reduces to [itex]2\pi^2/3[/itex]. Dividing both sides by four gives us:

  4. Feb 24, 2005 #3
    I see clearly now!--thanks

    (Will no one answer my "digit-factorial question" thread :frown: )
    Last edited: Feb 24, 2005
  5. Feb 24, 2005 #4


    User Avatar
    Science Advisor
    Homework Helper

    And let's not forget Euler's original method.Combining the series he found for [itex] \frac{\sin x}{x} [/itex] and the one from Taylor expansion,he was able to prove it...

  6. Feb 24, 2005 #5

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    And there's another way:


    Evalutate that as x and y both go from 0 to 1. Do it using a substitution, and then do it by replacing the fraction inside with its series expansion and ignore the convergence issues to rearrange sum and integral.
  7. Sep 1, 2011 #6
    So this is parsevals theorem?

    [tex] \frac{1}{\pi}\int_{-\pi}^\pi f^2(x)\:dx=\frac{a_0^2}{2}+\sum_{k=1}^\infty\left( a_k^2+b_k^2) [/tex]

  8. Sep 1, 2011 #7


    Staff: Mentor

    How about this?

    [tex] \frac{1}{\pi}\int_{-\pi}^\pi f^2(x)\:dx=\frac{a_0^2}{2}+\sum_{k=1}^\infty( a_k^2+b_k^2) [/tex]
  9. Sep 10, 2012 #8
    hello! i am new to this forum and it looks like a very nice place!

    sorry for my english, i hope everyone can understand it..

    sorry for spamming in this thread but it looks like it is the most close to what i need.

    i think i understood the answer master_coda gave but i don't understand why he choose f(x)=x..

    for instance in my exercise i am asked to verify this equation


    what f(x) should i choose for the calculation?
  10. Sep 10, 2012 #9

    I'm answering this, in spite of being an intent of "kidnapping" a thread because

    (1) it is, perhaps unwillingly, very close to the OP, and more important

    (2) This is a newcomer so he/she doesn't know (but now you do!).

    Check the following:

    $$\frac{\pi^2}{8}=\sum_{n=0}^\infty\frac{1}{(2n+1)^2}\Longleftrightarrow \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$$

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Convergence to pi^2/6