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theorem
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Homework Statement
Evaluate following series:
[tex]\sum_{n=1}^\infty \frac{1}{(4n^2-9)^2} [/tex]
by finding the Fourier series for the [itex]2\pi[/itex]-periodic function
[tex]f(x) =
\begin{cases}
sin(3x/2) & 0<x<\pi \\
0 & otherwise
\end{cases}
[/tex]
Homework Equations
[tex]a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)cos(nx)dx = -\frac{6}{\pi(4n^2-9)} [/tex]
[tex]b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)sin(nx)dx = \frac{4ncos(\pi n)}{\pi(4n^2-9)} [/tex]
[tex]f(x) = \frac{1}{2}a_0 + \sum_{n=1}^\infty \left(a_ncos(nx)+b_nsin(nx)\right) \\ = \frac{1}{2}\frac{2}{3\pi} + \frac{1}{\pi}\sum_{n=1}^\infty \left( \frac{4ncos(\pi n)sin(nx)-6cos(nx)}{4n^2-9}\right)[/tex]
The Attempt at a Solution
I have found the Fourier series and evaluated 1/(4n^2-9) as it was the first part of this exercise. However, I am not sure how to evaluate it for the square. I figured I just needed to square the answer but it turns out it's that simple .
I found Parseval's formula in my book
[tex] \frac{1}{\pi}\int_T |f(x)|^2dx = \frac{1}{2}|a_0|^2 + \sum_{n=1}^\infty (|a_n|^2+|b_n|^2) [/tex].
which I tried using, but the [itex]b_n[/itex] term gives me an ugly expression in the numerator.
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