There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required.
$$\begin{align} \sum_{n=0}^{\infty} \frac1{n!} = e \\ \lim_{n \to \infty} \left(1 + \frac1n \right)^n = e \\ \lim_{n \to \infty} \left(1 - \frac1n \right)^n = \frac1e \\ \lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} = e \\ \lim_{n \to \infty} \frac{1}{n} = 0 \\ \sum_{n=0}^{\infty} \frac1{n} \text{Diverges.} \end{align}$$