> [!NOTE] Definition (Real Exponential Function as a Power Series) > The [[Real Exponential Function|real exponential function]] can be defined as the [[Power Series|univariate real power series about zero]], $\exp(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n!}$ where $x\in \mathbb{R}$ is a variable; $x^{n}$ denotes an [[Integer Power of Real Number|integer power]] and $n!$ denotes the [[Factorial|nth factorial]]. # Properties By [[Radius of Convergence of Real Exponential Function Power Series]], for all $x\in \mathbb{R},$ $\sum_{n=0}^{\infty} \frac{x^{n}}{n! }<\infty.$ By [[Equivalence of Real Exponential Function as Limit of a Sequence and Power Series]], $\sum_{0}^{\infty} \frac{x^{n}}{n!} =\lim_{ n \to \infty }\left( 1-\frac{x}{n} \right)^{n}.$