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

Question

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?

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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.

Answer 1

When $n>k$, $a_n=0$. So the series is actually a polynomial.