# To find Radius Of Convergence of Power series

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.

• You did not apply Cauchy-Hadamard correctly. $\limsup a_n$ does not exist (indeed, what ordering you are using on $\mathbb{C}$ to talk about $\limsup a_n$?) Oct 13, 2023 at 18:36
• Are you assuming anything about $L$? Oct 13, 2023 at 18:37
• @José Carlos Santos Nothing . L is just a real number . Oct 13, 2023 at 18:41
• A real number? Why? Oct 13, 2023 at 18:41
• Sorry I mean a complex number Oct 13, 2023 at 18:44