List of Common or Useful Limits of Sequences and Series 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.
$$\begin{align}
\sum_{n=0}^{\infty} \frac1{n!} = e
\\ \lim_{n \to \infty} \left(1 + \frac1n \right)^n = e
\\ \lim_{n \to \infty} \left(1 - \frac1n \right)^n = \frac1e
\\ \lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} = e
\\ \lim_{n \to \infty} \frac{1}{n} = 0
\\ \sum_{n=0}^{\infty} \frac1{n} \text{Diverges.}
\end{align}$$

My opinion.
The most useful series is the geometric series, in both its finite and infinite forms:
$$
1 + x + x^2 + \cdots +  x^n = \frac{1 - x^{n+1}}{1-x} \quad (x \neq 1)
$$
and 
$$
1 + x + x^2 + \cdots   = \frac{1 }{1-x} \quad (|x| < 1)
$$
You can derive many others from it by substituting values and by formal manipulations (including differentiation and integration). It's as handy in combinatorics (as a generating function) as in analysis.
