Radius of convergence of the complex serie $(z-i)^n/n!$

I'm trying to find the radius of convergence of the following complex serie

$$\sum^\infty_{n=0} \frac{(z-i)^n}{n!}$$

I need to apply the ratio test, by evaluating $\lim_{n\rightarrow\infty}\left|(a_{n+1}/a_n)\right|$ but I don't know if I should consider the original expression as $a_n$ or if I need to factorize $z^n$ from the expression.

Both ways will work. If $a_n=\frac1{n!}$, then $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=0$, and therefore the radius of convergence is $\infty$.

Also,$$\lim_{n\to\infty}\left|\frac{\frac1{(n+1)!}(z-i)^{n+1}}{\frac1{n!}(z-i)^n}\right|=\lim_{n\to\infty}\frac{|z-i|}{n+1}=0.$$From this, it follows again that the radius of convergence is $\infty$.