Euler's Number: Limit of Sequence implies Limit of Series

Theorem
Let Euler's number $e$ be defined as:


 * $\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$

Then:


 * $\ds e = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$

That is:


 * $e = \dfrac 1 {0!} + \dfrac 1 {1!} + \dfrac 1 {2!} + \dfrac 1 {3!} + \dfrac 1 {4!} \cdots$