- #1

FeDeX_LaTeX

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## Homework Statement

(a) On (-π,π), find the Fourier series of f(x) = x.

(b) Hence, or otherwise, find the Fourier series of g(x) = x

^{2}

(c) Hence, show that [tex]\sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}[/tex]

## Homework Equations

[tex]f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{L} + b_n \sin \frac{n \pi x}{L} \right)[/tex]

where

[tex]a_0 = \frac{1}{L} \int_{-L}^{L} f(x) dx[/tex]

[tex]a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \left( \frac{n \pi x}{L} \right) dx[/tex]

[tex]b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \left( \frac{n \pi x}{L} \right) dx[/tex]

## The Attempt at a Solution

I found (a) as

[tex]x = \sum_{n=1}^{\infty} \frac{2}{n} (-1)^{n+1} \sin(nx)[/tex]

but for (b), how do I obtain the Fourier series for x

^{2}, without starting from scratch? I can derive that Fourier series fine and I use Parseval's identity for the last part and the result follows -- but I'm just puzzled about the 'hence' part of question (b). Are you supposed to integrate both sides from 0 to x? Doing that doesn't seem to give me the Fourier series for x

^{2}...