The real exponential function and the number $e$ by a power series

For $x\in \mathbb{R}$, define $\exp (x):={\sum }_{n=0}^{\mathrm{\infty }}\frac{x^n}{\iota (n!)},e:=\exp (1),$ provided by the all-real convergence proved in The exponential series converges absolutely for every real argument. Here $n!\in \mathbb{N}$ is the factorial of The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$, $\iota (n!)$ is its nonzero real image (The canonical natural $\iota (n)=n\cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing), and powers and series are those of Integer powers $a^m$ and Series, partial sums, convergence and the sum, divergence, and the tail series.

This is a real power series centred at $0$ (A real power series about a centre, its interval of convergence, and its radius in $[0,+\mathrm{\infty }]$). No logarithm, irrational power, or differential equation enters the definition.

The exponential series converges absolutely for every real argument ↗

The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$

The canonical natural $\iota (n)=n\cdot 1_F$ of a field

Canonical naturals are positive and strictly increasing

Integer powers $a^m$

Series, partial sums, convergence and the sum, divergence, and the tail series

A real power series about a centre, its interval of convergence, and its radius in $[0,+\mathrm{\infty }]$