Exponential of a complex number converges absolutely

Question

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$

This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?

Answer 1

The ratio from one term to the next is $z/n$. Eventually, $n$ is bigger than $|z|$, so the terms from that point on get smaller. Eventually, $n>2|z|$, so the terms halve in size each step. So the sum will converge. That is true for any $z$.
We don't have the problem of $\frac1{1-z}=1+z+z^2+z^3+...$ which stops converging when $|z|\geq1$.