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Math Help - Fourier series of a function

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    MHF Contributor arbolis's Avatar
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    Fourier series of a function

    The exercise states "Let f(x) be a periodic function with period 2 \pi defined in the interval [- \pi , \pi] by f(x)=1 if 0 \leq x \leq \pi and f(x)=0 if -\pi \leq x <0.
    Calculate the Fourier coefficients of f and obtain its Fourier series and analyze the convergence of the series.
    My attempt: First of all I think they made an error and meant to say the interval [- \pi , \pi). Otherwise it's not making sense to me.

    Ok so I read the definition of the nth coefficient of the Fourier series in my class notes and I read \hat f(n)= \frac{1}{2 \pi} \int _{- \pi}^{\pi} f(\theta ) e^{-in\theta } d\theta.
    So I calculated \hat f(0) to be worth \frac{1}{2}.
    For all n even and n \neq 0, I found out that \hat f(n)=0. While for all n uneven, I found out that \hat f(n)=-\frac{1}{\pi n}.
    Since the Fourier series is defined as \sum _{- \infty}^{\infty} \hat f(n) e^{in\theta}, I reach that the Fourier series is worth \frac{1}{2}+ \sum _{n=-\infty}^{\infty} \frac{- \cos (n \theta)+ i \sin (n \theta)}{n \pi} where n=2k+1 with k \in \mathbb{Z}, n\neq 0.
    To check out the convergence of the series, I guess I should split the series in 2, starting from -\infty to 0 and from 0 to +\infty, and then apply the quotient test?
    Is what I've done right? I'm not confident but can't find any error.

    Edit: I forgot to multiply by "i" for the series of the uneven terms I think. I don't think the result change that much, except with an "i" factor.
    Last edited by arbolis; July 27th 2010 at 10:07 AM.
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