Hello I have this question, and was wondering if anybody could help with it.

"Find the radius of convergence of the power series

$\displaystyle 1 +az + \frac{a(a+1)}{2!}z^2 + \cdots + \frac{a(a+1)\ldots (a+n-1)}{n!}z^n + \cdots $,

where a is a fixed real number (the answer will depend on a)."

However isnt that power series just $\displaystyle \sum_{n=0}^{\infty} \frac{ (a+n-1)!}{n!(a-1)!} z^n = \sum_{n=0}^{\infty} {a+n-1 \choose{n}} z^n$, which has radius of convergence 1, which doesnt depend on a? Any help with this would be appreciated .