*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 arranged in a linear fashion; such that each member comes either before, or after, every other member, and the order of members is important.

For example, (C,Y,R) is a sequence of letters; the ordering is that C is first, Y is second, and R is third. Sequences can be *finite*, as in the example just given, or *infinite*, such as the sequence of all even positive integers (2,4,6,...). Finite sequences include the *null sequence* ( ) that has no elements. The elements in a sequence are also called *terms*, and the number of terms (possibly infinite) is called the *length* of the sequence.

A sequence is denoted (*a*_{1},*a*_{2}, ...). For shortness, the notation (*a*_{n}) is also used.

A more formal definition of a **finite sequence** with terms in a set *S* is a function from {1,2,...,*n*} to *S* for some *n*≥0. An **infinite sequence** in *S* is a function from {1,2,...} (the set of natural numbers) to *S*.

A finite sequence is also called an n-tuple. A function from all integers into a set is sometimes called a **bi-infinite sequence**, since it may be thought of as a sequence indexed by negative integers grafted onto a sequence indexed by positive integers.

## Types and properties of sequences

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

If the terms of the sequence are a subset of a ordered set, then a **monotonically increasing** sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called **strictly monotonically increasing**. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called **monotonic** or **monotone**. This is a special case of the more general notion of monotonic function.

If the terms of a sequence are integers, then the sequence is an **integer sequence**. If the terms of a sequence are polynomials, then the sequence is a **polynomial sequence**.

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

## Series

The sum of a sequence of real numbers is a series. Alternately stated, a series is a sequence of partial sums. For example:

## See also

## External link

The On-Line Encyclopedia of Integer Sequences

Last updated: 10-24-2005 23:34:30