# 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) arranged in a "linear" fashion, such that the order of the members is well defined and significant.

For example, (C,Y,R) is a sequence of letters that differs from (Y,C,R), as the ordering matters. Sequences can be *finite*, as in this example, or *infinite*, such as the sequence of all even positive integers (2,4,6,...).

The members of a sequence are also called its *elements* or *terms*, and the number of terms (possibly infinite) is called the *length* of the sequence.

## Contents

## Examples and notation

There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below.

A sequence may be 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.

Finite sequences include the *null sequence* ( ) that has no elements.

## 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.

The terms **non-decreasing** and **non-increasing** avoid any possible confusion with strictly increasing and strictly decreasing, respectively, see also strict.

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.

## Sequences in analysis

In analysis, when talking about sequences, one usually understands sequences of the form

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, x_2, x_3, ...)}**or**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0, x_1, x_2, ...)}**

i.e. infinite sequences of elements indexed by natural numbers.
(It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by *x _{n}* = 1/log(

*n*) would be defined only for

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\ge2}**. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N}**.)

The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

## Series

The sum of a sequence is a series. More precisely, if **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, x_2, x_3, ...)}**
is a sequence, one considers the sequence of partial sums **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S_1, S_2, S_3, ...)}**
with

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n=x_1+x_2+\dots + x_n=\sum\limits_{i=1}^{n}x_i.}**

This new sequence is called a *series* with the terms **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1, x_2, x_3, ...}**
and is denoted as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum\limits_{i=1}^{\infty}x_i.}**

If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see series.

## See also

- Cauchy sequence
- Farey sequence
- Thue-Morse sequence
- Fibonacci sequence
- Net (topology) (a generalization of sequences)
- Sequence space
- Arithmetic progression
- Geometric progression
- limit of a sequence

## External links

The On-Line Encyclopedia of Integer Sequences

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