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.

  • 1
    $\begingroup$ 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 $\limsup a_n$?) $\endgroup$ Oct 13, 2023 at 18:36
  • $\begingroup$ Are you assuming anything about $L$? $\endgroup$ Oct 13, 2023 at 18:37
  • $\begingroup$ @José Carlos Santos Nothing . L is just a real number . $\endgroup$ Oct 13, 2023 at 18:41
  • $\begingroup$ A real number? Why? $\endgroup$ Oct 13, 2023 at 18:41
  • $\begingroup$ Sorry I mean a complex number $\endgroup$ Oct 13, 2023 at 18:44


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