# Radius of convergence of $f(z)?$

Let $p(x)$ be a polynomial of the real variable $x$ of degree $k\geq 1$ .Consider the power series $$f(z)=\sum_{n=0}^{\infty}p(n)z^n$$ where z is a complex variable .Then the radius of convergence of $f(z)$ is ?

• Use Ratio Test or Root Test. – André Nicolas Apr 21 '16 at 2:20
• $(\limsup_{n\rightarrow \infty} |p(n)|^{1/n})^{-1}$ – Rick Sanchez Apr 21 '16 at 2:29
• what is the radius of convergence of $\sum_n n^k z^n$ ??? – reuns Apr 21 '16 at 2:40

$\lim_{n\rightarrow \infty} \frac{p(n+1)}{p(n)}=1$ for any polynomial of degree $k \ge 1$.Therefore radius of convergence is 1 for given power series.