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Math Help - Power series question

  1. #1
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    Power series question

    Let p be a polynomial of degree k>0. Prove that \sum p(n)z^n has radius of convergence 1 and that there exists a polynomial q(z) of degree k such that \sum_{n=0}^{\infty}p(n)z^n = q(z)(1-z)^{-(k+1)} for |z|<1.

    I have proven the first part using the ratio test easily enough. For the second part I wrote p(n)=a_0 + a_1n + {a_2}n^2+...+{a_k}n^k and tried to expand the sum but didn't get anywhere productive. Can anyone help?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Power series question

    Hint If f(z)=\sum_{n=0}^{+\infty}p(n)z^n\;(|z|<1) , then f(z)-zf(z)=p(0)+\sum_{n=1}^{+\infty}[p(n)-p(n-1)]z^n and q(n)=p(n)-p(n-1) has degree less or equal than p .


    Edited: Of course "less" instead of "less or equal".
    Last edited by FernandoRevilla; December 28th 2011 at 07:14 AM.
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