Question
Let $\sum_{n=0}^\infty a_nz^n$ be a convergent series such that $\lim_{n\to\infty} a_n = L$. Let $P(z)$ be a polynomial of degree $s$. Then what is the radius of convergence of series $\sum_{n=0}^\infty P(n)a_nz^n$.
My attempt
$\limsup_{n\to\infty} a_n = \frac{1}{R}$ where $R$ is the Radius of Convergence. Since limit exists $\limsup_{n\to\infty}a_n = \lim_{n\to\infty}a_n=L $. $\limsup (a_nP(n)) =L\limsup P(n)$
I don't know how to proceed after this. I am stuck on finding limit supremum of $P(n)$. I am not good in solving limit problems, so please apologize if my doubt is silly.