> [!NOTE] Definition (Real Exponential Function as a Power Series) > The 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]]. **Note**: By [[Radius of Convergence of Real Exponential Function Power Series]], for all $x\in \mathbb{R},$ $\sum_{n=0}^{\infty} \frac{x^{n}}{n! }<\infty.$ > [!NOTE] Definition (Real Exponential Function as Limit of a Sequence) > The real exponential function is defined as the following [[Convergence|limit of a sequence of real numbers]]: $\text{exp}(x)=\lim_{ n \to \infty }\left( 1-\frac{x}{n} \right)^{n}.$ **Note**: By [[Convergence of Real Exponential Function Sequence]], $\left( 1-\frac{x}{n} \right)^{n}$ is indeed convergent for all $x\in \mathbb{R}.$ Moreover, by [[Equivalence of Real Exponential Function as Limit of a Sequence and Power Series]], $\sum_{0}^{\infty} \frac{x^{n}}{n!} =\lim_{ n \to \infty }\left( 1-\frac{x}{n} \right)^{n}.$ # Properties By [[Real Exponential Function of Sum]], $\exp(x+y)=\exp(x)\exp(y).$ By [[Real Exponential Function is Strictly Increasing]], for all $x>y,$ $\exp(x)>\exp(y).$ By [[Inverse of Real Exponential Function Exists, is Strictly Increasing and Continuous]], the exponential function has a continuous, strictly increasing function inverse. Note that its inverse is known as the [[Real Natural Logarithm Function|real natural logarithm]]. By [[Derivative of Real Exponential Function]], $\frac{d}{dx}e^{x}= e^{x}.$ # Applications Let $x>0,$ $p\in \mathbb{R}.$ We define the $p$th [[Real Power of Real Number|power]] of $x$ by $x^{p}= \exp(p\log(x)).$ We define [[Euler's Number|Euler's number]] by $e= \exp(1).$ Thus $\exp(p)= e^{p}$ as shown in [[Real Exponential Function is Real Power of Euler's Number]].