$$\mathrm{e} = \lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{n}$$
$$\mathrm{e} = \sum_{n=0}^{\infty} \frac{1}{n!}$$
$$\mathrm{e}^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}$$