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

I have proven the first part using the ratio test easily enough. For the second part I wrote $\displaystyle p(n)=a_0 + a_1n + {a_2}n^2+...+{a_k}n^k$ and tried to expand the sum but didn't get anywhere productive. Can anyone help?