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Math Help - 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:

    \arctan 2x = \int_0^{2x} {\frac{1}<br />
{{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 \arctan2x. The next step is to multiply the sum (which you'll get) by 4x and there's your series.

    From there you can find its radius of convergence.
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