> [!NOTE] Definition (Real Exponential Function as Limit of a Sequence) > The [[Real Exponential Function|real exponential function]] is defined as the following [[Convergence|limit of a sequence of real numbers]]: $\text{exp}(x)=\lim_{ n \to \infty }\left( 1-\frac{x}{n} \right)^{n}.$ # Properties 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]], $\sum_{0}^{\infty} \frac{x^{n}}{n!} =\lim_{ n \to \infty }\left( 1-\frac{x}{n} \right)^{n}.$