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Thread: Power Series Assignment

  1. #1
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    Power Series Assignment

    So my calculus professor has given us this online assignment on power series, and I can't wrap my head around in for the life of me.
    Most of the questions are the same, so I think if I get the process down for one of them, then I'll be able to approach the other ones.

    ex. The function f(x)= 4xarctan(2x) is represented as a power series:
    (see attached image at the bottom, since I'm not sure on how to input that on the computer)


    Find the first few coefficients in the power series.
    c0=
    c1=
    c2=
    c3=
    c4=

    Find the radius of convergence R of the series.



    Any help would be greatly appreciated!!
    Attached Thumbnails Attached Thumbnails Power Series Assignment-b534f3268e78d4712f28434aff5f4b1.png  
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    You can find the series representation as follows:

    $\displaystyle \arctan 2x = \int_0^{2x} {\frac{1}
    {{1 + u^2 }}\,du} = \int_0^{2x} {\left\{ {\sum\limits_{k = 0}^\infty {( - 1)^k u^{2k} } } \right\}\,du} .$

    Now you just interchange the sum by the integral and you'll get the series for $\displaystyle \arctan2x.$ The next step is to multiply the sum (which you'll get) by $\displaystyle 4x$ and there's your series.

    From there you can find its radius of convergence.
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