# Sequence

*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 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 a set of polynomials, the sequence is a polynomial 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.

A *subsequence* of a sequence *S* is a sequence formed from *S* by deleting some of the elements without disturbing the relative positions of the remaining elements.

The sum of a sequence is a series. For example:

## See also