Sequence

A sequence is an ordered list of terms. Sequences may be either finite or infinite.

Definition

A sequence of real numbers is simply a function $f : \mathbb{N} \rightarrow \mathbb{R}$ . For instance, the function $f(x) = x^2$ defined on $\mathbb{N}$ corresponds to the sequence $X = (x_n) = (0, 1, 4, 9, 16, \ldots)$ .

Convergence

Let $(x_n)$ be a sequence of reals. $(x_n)$ converges to $L \in \mathbb{R}$ if and only if for all positive reals $\epsilon$ , there exists a positive integer $k$ such that for all integers $n \ge k$ , we have $|x_n - L| < \epsilon$ . If $(x_n)$ converges to $L$ , $L$ is called the limit of $(x_n)$ and is written $\lim_{n \to \infty} x_n$ .

Resources

Online Encyclopedia of Integer Sequences -- A really cool math tool.

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Definition

Convergence

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