Radius of convergence of $\sum a_nx^n$ where $a_n = {k \choose n}$

Consider the power series $\sum a_n x^n$ where

$$a_n = {k \choose n}$$

for some $k$. What is the radius of convergence of this power series? I got one. Does that seem correct?

I got that the radius of convergence is one as follows: In Ross' text, we define the radius of convergence as $1/\beta$ where $\beta = \lim\left|a_{n+1} / a_n \right|$. I just considered

$$\lim\left|{k \choose n+1} \cdot {k \choose n}^{-1} \right|$$

and working out the algebra, got $\beta = 1$ so the radius of convergence is one.

When $n>k$, $a_n=0$. So the series is actually a polynomial.
• So is the radius of convergence not one? I used the standard $\lim\left|a_{n+1}/a_n\right|$ to get one – Jordan Kelk Apr 9 '14 at 3:03