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Math Help - Power Series problem

  1. #1
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    California
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    [Solved] Power Series problem

    Hi,
    This math problem is really confusing me, any ideas on how to start?

    i have to explicitely compute the function f defined by

    f(x)= \sum_{n=1}^{\infty} n^2x^_2n}

    as well as find its domain.

    from wolframalpha i know the correct function is f(x)=\frac{-x^2(x^2+1)}{(x^2-1)^3} although i have no idea how to prove this and it seems quite complex for a power series.

    Any help is greatly appreciated.
    Last edited by Calculon; May 2nd 2012 at 10:11 PM. Reason: Solved
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  2. #2
    Junior Member ignite's Avatar
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    Kanpur,India
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    Re: Power Series problem

    t=x^2 (say)
    \sum_{n=0}^{\infty} t^n = \frac{1}{1-t}
    Differentiate wrt t,
    \sum_{n=0}^{\infty} nt^{n-1} = \frac{1}{(1-t)^2}
    \sum_{n=0}^{\infty} nt^{n} = \frac{t}{(1-t)^2}
    Differentiate wrt t,
    \sum_{n=0}^{\infty} n^2t^{n-1} = \frac{1+t}{(1-t)^3}
    \sum_{n=0}^{\infty} n^2t^{n} = \frac{t(1+t)}{(1-t)^3}
    For n=0,the term itself is 0.So,
    \sum_{n=1}^{\infty} n^2t^{n} = \frac{t(1+t)}{(1-t)^3}

    Now Substitute back t=x^2.
    To look for the domain check in which region your series converges.
    Thanks from Calculon
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  3. #3
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    Re: Power Series problem

    Thanks alot!
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