# Exponential of a complex number converges absolutely

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

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

• Can you define a layman? Is it someone with no mathematics background at all? Feb 14, 2015 at 12:59
• What would a layman need complex numbers, exponentials, and power-series for? Feb 14, 2015 at 12:59
• Hmmm. I have done heaps of math, but I don't know what converging absolutely means, especially on complex numbers Feb 14, 2015 at 13:00
• The layman is me if you want to just have a quick look at my past questions Feb 14, 2015 at 13:00
• Are you asking the definition of 'absolute convergence'? Feb 14, 2015 at 13:02

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 $|z|\geq1$.