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Need help with a sum series (Frobenius series)

  1. Feb 1, 2012 #1
    1. The problem statement, all variables and given/known data
    I've obtained a recurrence relation of:
    [itex]a_{n+2}[/itex] = [itex]\frac{(n-1)(n-2)-\frac{2k}{w_{o}^{2}}}{R^{2}(n+1)(n+2)}[/itex][itex]a_{n}[/itex]
    from a Frobenius series solution problem and I've expanded it to give the series:

    f(r) = 1 + [itex]\frac{-\frac{2k}{w_{o}^{2}}}{6R^{2}}[/itex][itex]r^{2}[/itex] + [itex]\frac{-\frac{20k}{w_{o}^{2}}+\frac{4k^{2}}{w_{o}^{4}}}{120R^{4}}[/itex][itex]r^{4}[/itex] + [itex]\frac{-\frac{560k}{w_{o}^{2}}+\frac{152k^{2}}{w_{o}^{4}}-\frac{8k^{3}}{w_{o}^{6}}}{5040R^{6}}[/itex][itex]r^{6}[/itex] ...

    I can see that the denominator is 3![itex]R^{2}[/itex] , 5![itex]R^{4}[/itex] , 7![itex]R^{6}[/itex] which is (2n+1)! [itex]R^{2n}[/itex] however I need a bit of help condensing the numerator into a single series term.
  2. jcsd
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