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Help regarding a question Parseval's Theorem

  1. Oct 29, 2008 #1

    TFM

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    1. The problem statement, all variables and given/known data

    The Fourier series for f(x) = x2 over the interval (−1/2, 1/2) is:

    [tex] f(x) = \frac{1}{12}-\frac{1}{\pi^2} (cos 2\pi x - \frac{1}{2^2}cos4\pi x + \frac{1}{3^2}cos6\pi x) ... [/tex]

    Using Parseval's Theorem, show that

    [tex] \sum _{n = 1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90} [/tex]

    2. Relevant equations

    Fourier's Series:

    [tex] f(x) = \frac{1}{2}a_0 + \sum _{n = 1}^\infty a_n cos\frac{2\pi nx}{l} \sum _{n = 1}^\infty b_n sin \frac{2\pi nx}{l}[/tex]

    [tex] a_0 = \frac{1}{l} \int^{\frac{l}{2}}_{\frac{-l}{2}}f(x) dx [/tex]

    [tex] a_n = \frac{1}{l} \int^{\frac{l}{2}}_{\frac{-l}{2}}f(x) cos \frac{2\pi nx}{l} dx [/tex]


    [tex] b_n = \frac{1}{l} \int^{\frac{l}{2}}_{\frac{-l}{2}}f(x) sin \frac{2\pi nx}{l} dx [/tex]

    Parseval's Theorem:

    [tex] \frac{1}{2\pi} \int^{\pi}_{-\pi}f(x)^2 dx = \frac{1}{4}a_0^2 + \frac{1}{2} \sum _{n = 1}^\infty a_n^2 + \frac{1}{2}\sum _{n = 1}^\infty b_n^2 [/tex]

    3. The attempt at a solution

    See I'm not quite sure where to go from here. It says that it is a Fourier series, but it doesn't seem to fit with the definition of Fourier Series I have quoted below?

    Any assitance would be greatly appreciated,

    TFM
     
  2. jcsd
  3. Oct 29, 2008 #2

    Office_Shredder

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    You multiplied the cosine and sine portions of the Fourier series together, they're supposed to be added. At any rate

    For your f(x), a0=1/6 and then you can distribute the [tex]\frac{-1}{\pi^2}[/tex] to calculate the other ai's

    And then the sine component is just all zeroes (so all your bi's will be 0)
     
  4. Oct 29, 2008 #3

    TFM

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    So firstly,

    [tex] f(x) = \frac{1}{2}a_0 + \sum _{n = 1}^\infty a_n cos\frac{2\pi nx}{l} + \sum _{n = 1}^\infty b_n sin \frac{2\pi nx}{l} [/tex]

    And

    [tex] a_0 = \frac{1}{6} [/tex]

    As there are no sins, this implies that b_n = 0 (since b_n is the sin component.)
    Multiplying out the brackets:

    [tex] f(x) = \frac{1}{12} - (\frac{1}{\pi^2}cos 2\pi x - \frac{1}{\pi^2}\frac{1}{2^2}cos4\pi x + \frac{1}{\pi^2}\frac{1}{3^2}cos6\pi x ...) [/tex]

    [tex] f(x) = \frac{1}{12} - (\frac{1}{\pi^2}cos 2\pi x - \frac{1}{4\pi^2}cos4\pi x + \frac{1}{9\pi^2}cos6\pi x ...) [/tex]

    So the a_n seems to be an increasing value:

    [tex] a_n = frac{1}{\pi^2}, frac{1}{4\pi^2}, frac{1}{9\pi^2}... [/tex]

    Does this seem okay so far?

    ???

    TFM
     
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