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Math Help - Fourier Series

  1. #1
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    Fourier Series

    I have to say that, perhaps it's just very late in the semester, but I have no idea what's going on. I'm asked to show that x = \pi - 2 \displaystyle \sum_{n = 1}^{\infty} \frac{sin(nx)}{n} for 0 < x < 2 \pi. The professor even started to answer this question in class, writing out the Fourier series of f(x) = x, but wasn't able to go through it all, and I didn't really understand even what he was doing, even after talking to him after class. Maybe it's a little bit of a language barrier...

    An answer to the question would be nice, but even better would be some clear statement of just how any such equality relates to the study of Fourier series would be better.
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  2. #2
    Behold, the power of SARDINES!
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    To calculate the Fourier coefficients use

    \displaystyle a_0=\frac{1}{\pi}\int_{0}^{2\pi}xdx=2\pi

    \displaystyle a_n=\frac{1}{\pi}\int_{0}^{2\pi}x\cos(nx)dx=0

    \displaystyle b_n=\frac{1}{\pi}\int_{0}^{2\pi}x\sin(nx)dx=-\frac{2}{n}

    Now we can write f as

    \displaystyle f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(nx)+  b_{n}\sin(nx)

    This gives

    \displaystyle x=\pi-2\sum_{n=1}^{\infty}\frac{\sin(nx)}{n}
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