The Convergence of a Complex Valued Infinite Series

Check the convergence and calculate the radius of convergence of the series
$$
\sum^{\infty}_{n=1}\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}z^n,\forall\alpha\in\mathbb{C}.

$$

I tried to use the Triangle Inequalities & Comparison Test but could not get the right answer. Could somebody help me with this please?

Answer

Let

$$u_n(z)=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}z^n$$
then by the ratio test we have

$$\left|\frac{u_{n+1}(z)}{u_n(z)}\right|=\frac{|\alpha-n|}{n+1}|z|\xrightarrow{n\to\infty}|z|$$

hence the series has the radius $R=1$.