# Math Help - Power series question

1. ## 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?

2. ## 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".