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Finding order of an infinite series

  1. Mar 30, 2014 #1
    1. The problem statement, all variables and given/known data

    I have been working on a truncated Fourier series. I have come up with a truncated series for cos (αx) and it matches my book, where in this case I'm letting x=π, and then I have shown as the book asks,

    Truncated series = [itex]F_{N}(π)[/itex] = cos (απ) + [itex]\frac{2α}{π} \sin{(απ)}[/itex][itex]\sum\limits_{k=N+1}^\infty[/itex][itex]\frac{1}{k^2 - α^2}[/itex]

    I am then asked to show that this approximates to almost the same thing:

    [itex]F_{N}(π)[/itex] = cos (απ) + [itex]\frac{2α}{Nπ} \sin{(απ)}[/itex]

    So I thought no worries, looks like I need to take that infinite sum there and simply show that it is of order 1/N.

    2. Relevant equations



    3. The attempt at a solution


    The only way I know how to try that is to pretend the sum is actually an integral,

    [itex] \int \limits_{N+1}^\infty[/itex][itex]dk \frac{1}{k^2 - α^2}[/itex], and I wound up with something like:

    [itex] \frac{\frac{αk}{α^2 - k^2} + arctan \frac{k}{α} }{2α^3}[/itex]

    as the solution to the integral, with limits of k=infinity and k=N+1 in there for the definite integral (I can't figure out how to show that - I was lucky to manage what I have.

    Anyway, having done that, it really doesn't look like it's of order 1/N to me. Am I heading the right way with this, or is there some other way to establish that it's of order 1/N?

    This is the very last step in the question and I get the idea it's meant to be, if not trivial, at least quite an easy part.
     
    Last edited: Mar 30, 2014
  2. jcsd
  3. Mar 31, 2014 #2

    Mark44

    Staff: Mentor

    I don't know if your result is correct. An easier way to do the integration is to decompose the integrand using partial fractions.

    ##\frac{1}{k^2 - α^2} = \frac{A}{k - α} + \frac{B}{k + α}##
    Solve the equation above for A and B and then integrate.
     
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