The Real Exponential Function

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let convergence and sums of series be as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence, and for $u\in\mathbb{R}$ and $k\in\mathbb{N}$ let the powers $u^{k-1}$, the factorials $(k-1)!$ and the quotients $u^{k-1}/(k-1)!$ be as in The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ.

(i) For $u\in\mathbb{R}$, $\sum_{k=0}^{\infty}\frac{u^{k}}{k!}$ denotes the series $\sum_{k=1}^{\infty}\frac{u^{k-1}}{(k-1)!}$ of the sequence of The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ (i), and also its sum, which exists as the series converges absolutely, and hence converges, by The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ (ii).

(ii) The exponential function is the map

$$\exp:\mathbb{R}\to\mathbb{R},\qquad u\mapsto\exp(u)=\sum_{k=0}^{\infty}\frac{u^{k}}{k!},$$

of The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ (iii).

(iii) The number $e$ is $e=\exp(1)$.