### Notation $e^x = \exp(x)$ ### Definitions $\begin{aligned} e^{x}&=\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}\\ &=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\cdots\\ &=1+\sum_{n=1}^{\infty}\left(\prod_{i=1}^{n} \frac{x}{i}\right) \\&=\frac{1}{1-\frac{x}{1+x-\frac{\frac{1}{2} x}{1+\frac{1}{2} x-\frac{\frac{1}{3} x}{1+\frac{1}{3} x-\frac{\frac{1}{4} x}{1+\frac{1}{4} x-\ddots}}}}}\\ &=\frac{1}{1-\frac{x}{1+x-\frac{2 x}{2+x-\frac{2 x}{3+x-\frac{3 x}{4+x-\ddots}}}}} \end{aligned}$ https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function from [[𝑒 - Euler's Number]]