*This is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical).*

In mathematics, a **sequence** is a list of objects (or events) which have been ordered in a numerical (and sequential) fashion; such that each member either comes before, or after, every other member. A sequence is a function with a domain equal to the set of positive integers.

The sequence of positive integers is: 1, 2, 3, ..., *n* - 1, *n*, *n* + 1, ... Each number is a term, with *n* being the "*n*-th term". A sequence can be denoted by: **{ a_{n} }**; such that, in the above list of positive integers,

*a*

_{1}is 1,

*a*

_{317}is 317, and

*a*

_{n}is

*n*-- this is also indicated by:

*a*

_{0},

*a*

_{1},

*a*

_{2}, ...,

*a*

_{n}, ... The terms of a sequence, are part of a set, commonly indicated by

**S**; they are a "

**sequence in S**".

A sequence may have a *finite* or *infinite* number of terms; thus, it is called either *finite* or *infinite*. Obviously, it is impossible to give *all* the terms of an infinite sequence. Infinite sequences are given by listing the first few terms, followed by an ellipsis.

Formally, a sequence can be defined as a function from **N** (the set of natural numbers) into some set *S*.

If *S* is the set of integers, then the sequence is an integer sequence.

If *S* is endowed with a topology then it is possible to talk about **convergence** of the sequence. This is discussed in detail in the article about limits.

For a given sequence the corresponding sequence of partial sums is called an infinite series.

E.g.: 1 + 1/2 + 1/4 + ... is a convergent series, meaning that the sequence 1, 1 + 1/2, 1 + 1/2 + 1/4, ... is convergent.

A *subsequence* is a sequence with some of its members omitted.

See also: Farey sequence