Limit

The limit of a function $f(x)$ as $x$ approaches $a$ is $L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$. Symbolically, this is written as:

In this case, we write:

Sequence

A sequence is a function from the natural numbers to the real numbers. It is usually denoted as $(a_n)$, where $n$ is a natural number.

Alternative notations:

• $(a_1, a_2, a_3, \ldots)$
• $(a_n \colon n = 1, 2, 3, \ldots)$
• $(a_n)_{n=1}^{\infty}$

When each term of the sequence satisfies a certain condition P, we write:

Convergence & Divergence of sequences

When a sequence $(a_n)$ converges to a real number $L$, we write:

This means that for every $\epsilon > 0$, there exists a natural number $N$ such that if $n > N$, then $|a_n - L| < \epsilon$. Symbollcally, this is written as:

When a sequence $(a_n)$ diverges, we write:

Series

A series is a sequence of partial sums of a sequence. Given a sequence $(a_n)$, a series derived from it, is usually denoted as:

Or simply: