The Complex Exponential Series Converges Absolutely, and Its Sum Is a Function on ℂ

In the setting of Complex Analysis: Standing Notation, let convergence and sums and absolute convergence of series in $\mathbb{C}$ be as in Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence. For $z\in\mathbb{C}$ and $k\in\mathbb{N}$, $k-1\in\mathbb{N}_{0}$ is the difference, $z^{k-1}$ is the power, with $z^{0}=1$, and the factorial $(k-1)!$ is read in $\mathbb{C}$.

(i) The terms. For every $z\in\mathbb{C}$ there is exactly one sequence in $\mathbb{C}$ whose $k$-th term is $z^{k-1}/(k-1)!$ for every $k\in\mathbb{N}$.

(ii) Absolute convergence. For every $z\in\mathbb{C}$ the series

$$\sum_{k=1}^{\infty}\frac{z^{k-1}}{(k-1)!}$$

of the sequence of (i) converges absolutely, and in particular converges.

(iii) The sum as a function. There is exactly one function $E:\mathbb{C}\to\mathbb{C}$ with $E(z)=\sum_{k=1}^{\infty}\frac{z^{k-1}}{(k-1)!}$ for every $z\in\mathbb{C}$.