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.
$$\begin{align}
\sum_{n=0}^{\infty} \frac1{n!} = e
\\ \lim_{n \to \infty} \left(1 + \frac1n \right)^n = e
\\ \lim_{n \to \infty} \left(1 – \frac1n \right)^n = \frac1e
\\ \lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} = e
\\ \lim_{n \to \infty} \frac{1}{n} = 0
\\ \sum_{n=0}^{\infty} \frac1{n} \text{Diverges.}
\end{align}$$
Best Answer
This sheet by Dave Renfro that I found online was beyond helpful! http://mathforum.org/kb/servlet/JiveServlet/download/206-1874348-6544585-538002/seq3.pdf