Sequence

A sequence is an ordered list of terms. Sequences may be either finite or infinite. In mathematics we are often interested in sequences with specific properties, the Fibonacci sequence is perhaps the most famous example.

Definition

A sequence of real numbers is simply a function $f : \mathbb{N} \rightarrow \mathbb{R}$ . For instance, the function $f(x) = x^2$ 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 $\forall \epsilon > 0, \exists k \in \mathbb{N} : \forall n \in \mathbb{N}, n \ge k, |x_n - L| < \epsilon$ . 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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