- #1
TFM
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Homework Statement
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]
Homework 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]
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