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Fourier Series question

  1. Feb 20, 2008 #1
    Hey everyone,
    I got the following fourier series

    F.S f(x)= (pi/2) - (4/pi) [tex]\sum[/tex]n=1,3.. to infinity (1/n^2 cos (nx))

    l= pi

    After deriving it the question now is how can i use it to show

    [tex]\sum[/tex] n=1 to infinity (1/(2n-1)^2= 1+ 1/3^2 + 1/5^2 +... = pi^2/8

    I think I am not sure what I have to do here.

  2. jcsd
  3. Feb 20, 2008 #2
    Usually the idea is to find some value of x that makes your Fourier series into the series in n that you want. If you choose x=pi/2 for example, cos(nx) will vanish for odd n. Then you'd plug your choice of x into the function f(x), for which you presumably have a closed form (but you haven't told us what it is).
    Last edited: Feb 20, 2008
  4. Feb 20, 2008 #3


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    Can you think of a value for x such that the cosine becomes zero for all even n, and non-zero (for example 1) for all odd n? Then calculate f(x) for this x.
  5. Feb 20, 2008 #4
    f(x)= |x| over (-pi,pi]
    -x over -pi<x<= 0 and x over 0<x<=pi
  6. Feb 20, 2008 #5
    If i use x=pi/2 in fourier series cos nx vanishes and i have 1/n^2 which for odd numbers is 1/(2n-1)^2

    I can get that. how would i prove the part series converges to pi^2/8

  7. Feb 20, 2008 #6
    Forgot to mention the question. question was to use fourier series and x=0 to prove it. so i cannot use x= pi/2 can i?
  8. Feb 20, 2008 #7
    The x=pi/2 was just an example. It seems like the question even gives you the value of x you should use, so just plug it into the series and the function and you have your answer.
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