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Math Help - Question about Fourier Series

  1. #1
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    Question about Fourier Series

    Hi

    The following question i am unable to understand why the answer is:


    Because this is what i got:

    f(t) ~ \frac{1}{\pi} + \frac{sin(t)}{2} - \frac{2}{\pi} \sum\limits_{n=1}^{\infty}\frac{cos(nt)}{n^2-1}

    Hope someone can help me
    P.S
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  2. #2
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    Quote Originally Posted by Paymemoney View Post
    Hi

    The following question i am unable to understand why the answer is:


    Because this is what i got:

    f(t) ~ \frac{1}{\pi} + \frac{sin(t)}{2} - \frac{2}{\pi} \sum\limits_{n=1}^{\infty}\frac{cos(nt)}{n^2-1}

    Hope someone can help me
    P.S


    I get \displaystyle{a_n=\frac{1}{\pi}\int\limits^\pi_0 \sin t\cos nt\,dt=\frac{1}{\pi}\int\limits^\pi_0\left[\sin(n+1)t-\sin(n-1)t\right]dt=}

    \dispaystyle{\frac{1}{\pi}\left[-\frac{1}{n+1}\cos(n+1)t+\frac{1}{n-1}\cos(n-1)t\right]^\pi_0=}

    \displaystyle{=\frac{(-1)^{n+1}-1}{\pi}\cdot\frac{2}{n^2-1}=\left\{\begin{array}{cc}0&\mbox{, if }n\mbox{ is odd}\\-\frac{4}{\pi(n^2-1)}&\mbox{ , if }n\mbox{ is even}\end{array}\right.} , so I think both

    answers are wrong...or I am, of course.

    Tonio
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