The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let partial sums, convergence and sums of series and absolute convergence be as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence. For $k\in\mathbb{N}_{0}$ let $k!$ be the factorial of $k$, and for $u\in\mathbb{R}$ let $u^{k}$ be the power of $u$, with $u^{0}=1$ as in Powers with Exponents in the Natural Numbers with Zero (ii). Elements of $\mathbb{N}_{0}$ standing in real expressions are read in $\mathbb{R}$ by The Real Line: Standing Notation and Background for Calculus on Intervals (vii); read there, $1\le k!$, so $k!\neq0$, by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii). For $k\in\mathbb{N}$, $k-1\in\mathbb{N}_{0}$ is the difference, defined, since $1\le k$ by Arithmetic and Order of the Natural Numbers (x).

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

(ii) For every $u\in\mathbb{R}$, the series $\sum_{k=1}^{\infty}\frac{u^{k-1}}{(k-1)!}$ of the sequence of (i) converges absolutely, and in particular converges; its first term is $1$.

(iii) There is exactly one map $h:\mathbb{R}\to\mathbb{R}$ with $h(u)=\sum_{k=1}^{\infty}\frac{u^{k-1}}{(k-1)!}$ for every $u\in\mathbb{R}$.