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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.

$$\begin{align} \sum_{n=0}^{\infty} \frac1{n!} = e \\ \lim_{n \to \infty} \left(1 + \frac1n \right)^n = e \\ \lim_{n \to \infty} \left(1 - \frac1n \right)^n = \frac1e \\ \lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}} = e \\ \lim_{n \to \infty} \frac{1}{n} = 0 \\ \sum_{n=0}^{\infty} \frac1{n} \text{Diverges.} \end{align}$$

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    $\begingroup$ I would not try to remember a list of limits but rather learn a list of methods for determining the limits. E.G. the first three limits are all related to $\lim_{n\to \infty}(1+a/n)^n=e^a$. You get the first expression for $a=1$ and using the binomial expansion. $\endgroup$ – MrYouMath Jul 30 '16 at 8:07
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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

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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.

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