> [!NOTE] Lemma > The [[Radius of Convergence of Complex Power Series|radius of convergence]] of the [[Real Exponential Function|real exponential function power series]] is $\infty.$ **Proof**: The ratio of successive terms of the series is $\frac{x^{n+1}}{(n+1)!x^{n}} \frac{n!}{x^{n}}=\frac{x}{n+1}\to{0}$thus by [[Ratio Test for Series]], $\sum_{n=0}^{\infty} \frac{x^{n}}{n!}<\infty$ for all $x\in \mathbb{R}.$