$\displaystyle e = \lim_{n \rightarrow \infty} \left ( 1 + \frac{1}{n} \right )^n = \sum_{n=0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots$