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Power series

  1. Feb 5, 2005 #1
    Evaluate the indefinite integral as a power series and find radius of convergence. (i don't know how to type the integral and summations signs, sorry)

    (integral sign) (x-tan^-1x)/x^3 dx. ( if you write this out it makes more sense)

    i was able to find the power series of tan^-1x = x^(2n+1) (-1)^n/(2n+1).
    i don't know how to continue on with this. all we have learned is to use the power series of the geometric series 1/(1-x), and some integration/differentiation methods.

    i am rather confused on the whole topic, so if anyone has any ideas, the simplest explanations would be greatly appreciated. thanks
  2. jcsd
  3. Feb 5, 2005 #2


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    So your integral is
    [tex] \int \frac{x-\arctan x}{x^{3}} dx [/tex]

    ??Okay,for term by term integration of it,separate it into 2 integrals...Though it's not really helpful for the convergence part...

  4. Feb 5, 2005 #3
    that is the integral, thanks, but i am not sure what to do after pulling it apart into two integrals. quite honestly, i am puzzled by this whole topic. any advice?
  5. Feb 5, 2005 #4


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    1.Pull apart into integrals.
    2.Integrate the first.It's elementary.
    3.Write the series expansion of "artan" and devide its terms by x^{3}.
    4.Integrate by parts eery term of the new series...
    5.Think of a way to get the convergence radius.


    P.S.It can be done exactly (find the antiderivative).
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