# Sequence

"Sequential" redirects here. Sequence_sentence_0

For the manual transmission, see Sequential manual transmission. Sequence_sentence_1

For other uses, see Sequence (disambiguation). Sequence_sentence_2

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Sequence_sentence_3

Like a set, it contains members (also called elements, or terms). Sequence_sentence_4

The number of elements (possibly infinite) is called the length of the sequence. Sequence_sentence_5

Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Sequence_sentence_6

Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences), or the set of the first n natural numbers (for a sequence of finite length n). Sequence_sentence_7

For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. Sequence_sentence_8

This sequence differs from (A, R, M, Y). Sequence_sentence_9

Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequence_sentence_10

Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...). Sequence_sentence_11

In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. Sequence_sentence_12

The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context. Sequence_sentence_13

## Examples and notation Sequence_section_0

A sequence can be thought of as a list of elements with a particular order. Sequence_sentence_14

Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. Sequence_sentence_15

In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequence_sentence_16

Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers. Sequence_sentence_17

There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. Sequence_sentence_18

One way to specify a sequence is to list all its elements. Sequence_sentence_19

For example, the first four odd numbers form the sequence (1, 3, 5, 7). Sequence_sentence_20

This notation is used for infinite sequences as well. Sequence_sentence_21

For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Sequence_sentence_22

Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Sequence_sentence_23

Other ways of denoting a sequence are discussed after the examples. Sequence_sentence_24

### Examples Sequence_section_1

The prime numbers are the natural numbers bigger than 1 that have no divisors but 1 and themselves. Sequence_sentence_25

Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). Sequence_sentence_26

The prime numbers are widely used in mathematics, particularly in number theory where many results related to them exist. Sequence_sentence_27

The Fibonacci numbers comprise the integer sequence whose elements are the sum of the previous two elements. Sequence_sentence_28

The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). Sequence_sentence_29

Other examples of sequences include those made up of rational numbers, real numbers and complex numbers. Sequence_sentence_30

The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. Sequence_sentence_31

In fact, every real number can be written as the limit of a sequence of rational numbers (e.g. via its decimal expansion). Sequence_sentence_32

As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. Sequence_sentence_33

A related sequence is the sequence of decimal digits of π, that is, (3, 1, 4, 1, 5, 9, ...). Sequence_sentence_34

Unlike the preceding sequence, this sequence does not have any pattern that is easily discernible by inspection. Sequence_sentence_35

The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences. Sequence_sentence_36

### Indexing Sequence_section_2

In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. Sequence_sentence_37

In these cases the index set may be implied by a listing of the first few abstract elements. Sequence_sentence_38

For instance, the sequence of squares of odd numbers could be denoted in any of the following ways. Sequence_sentence_39

### Defining a sequence by recursion Sequence_section_3

Main article: Recurrence relation Sequence_sentence_40

Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion. Sequence_sentence_41

This is in contrast to the definition of sequences of elements as functions of their positions. Sequence_sentence_42

To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. Sequence_sentence_43

In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation. Sequence_sentence_44

The Fibonacci sequence is a simple classical example, defined by the recurrence relation Sequence_sentence_45

A complicated example of a sequence defined by a recurrence relation is Recamán's sequence, defined by the recurrence relation Sequence_sentence_46

A linear recurrence with constant coefficients is a recurrence relation of the form Sequence_sentence_47

A holonomic sequence is a sequence defined by a recurrence relation of the form Sequence_sentence_48

Not all sequences can be specified by a recurrence relation. Sequence_sentence_49

An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...). Sequence_sentence_50

## Formal definition and basic properties Sequence_section_4

There are many different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the definitions and notations introduced below. Sequence_sentence_51

### Definition Sequence_section_5

In this article, a sequence is formally defined as a function whose domain is an interval of integers. Sequence_sentence_52

This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). Sequence_sentence_53

However, many authors use a narrower definition by requiring the domain of a sequence to be the set of natural numbers. Sequence_sentence_54

This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Sequence_sentence_55

Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. Sequence_sentence_56

In some contexts, to shorten exposition, the codomain of the sequence is fixed by context, for example by requiring it to be the set R of real numbers, the set C of complex numbers, or a topological space. Sequence_sentence_57

Sequences and their limits (see below) are important concepts for studying topological spaces. Sequence_sentence_58

An important generalization of sequences is the concept of nets. Sequence_sentence_59

A net is a function from a (possibly uncountable) directed set to a topological space. Sequence_sentence_60

The notational conventions for sequences normally apply to nets as well. Sequence_sentence_61

### Finite and infinite Sequence_section_6

The length of a sequence is defined as the number of terms in the sequence. Sequence_sentence_63

A sequence of a finite length n is also called an n-tuple. Sequence_sentence_64

Finite sequences include the empty sequence ( ) that has no elements. Sequence_sentence_65

### Increasing and decreasing Sequence_section_7

The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively. Sequence_sentence_66

### Bounded Sequence_section_8

If the sequence of real numbers (an) is such that all the terms are less than some real number M, then the sequence is said to be bounded from above. Sequence_sentence_67

In other words, this means that there exists M such that for all n, an ≤ M. Any such M is called an upper bound. Sequence_sentence_68

Likewise, if, for some real m, an ≥ m for all n greater than some N, then the sequence is bounded from below and any such m is called a lower bound. Sequence_sentence_69

If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded. Sequence_sentence_70

### Subsequences Sequence_section_9

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. Sequence_sentence_71

For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of the positive integers (1, 2, 3, ...). Sequence_sentence_72

The positions of some elements change when other elements are deleted. Sequence_sentence_73

However, the relative positions are preserved. Sequence_sentence_74

### Other types of sequences Sequence_section_10

Some other types of sequences that are easy to define include: Sequence_sentence_75

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• An integer sequence is a sequence whose terms are integers.Sequence_item_0_0
• A polynomial sequence is a sequence whose terms are polynomials.Sequence_item_0_1
• A positive integer sequence is sometimes called multiplicative, if anm = an am for all pairs n, m such that n and m are coprime. In other instances, sequences are often called multiplicative, if an = na1 for all n. Moreover, a multiplicative Fibonacci sequence satisfies the recursion relation an = an−1 an−2.Sequence_item_0_2
• A binary sequence is a sequence whose terms have one of two discrete values, e.g. base 2 values (0,1,1,0, ...), a series of coin tosses (Heads/Tails) H,T,H,H,T, ..., the answers to a set of True or False questions (T, F, T, T, ...), and so on.Sequence_item_0_3

## Limits and convergence Sequence_section_11

Main article: Limit of a sequence Sequence_sentence_76

An important property of a sequence is convergence. Sequence_sentence_77

If a sequence converges, it converges to a particular value known as the limit. Sequence_sentence_78

If a sequence converges to some limit, then it is convergent. Sequence_sentence_79

A sequence that does not converge is divergent. Sequence_sentence_80

### Applications and important results Sequence_section_13

Moreover: Sequence_sentence_81

### Cauchy sequences Sequence_section_14

Main article: Cauchy sequence Sequence_sentence_82

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. Sequence_sentence_83

The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. Sequence_sentence_84

One particularly important result in real analysis is Cauchy characterization of convergence for sequences: Sequence_sentence_85

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• A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.Sequence_item_1_4

In contrast, there are Cauchy sequences of rational numbers that are not convergent in the rationals, e.g. the sequence defined by x1 = 1 and xn+1 = xn + 2/xn/2 is Cauchy, but has no rational limit, cf. Sequence_sentence_86

here. Sequence_sentence_87

More generally, any sequence of rational numbers that converges to an irrational number is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers. Sequence_sentence_88

Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called complete metric spaces and are particularly nice for analysis. Sequence_sentence_89

### Infinite limits Sequence_section_15

In this case we say that the sequence diverges, or that it converges to infinity. Sequence_sentence_90

An example of such a sequence is an = n. Sequence_sentence_91

and say that the sequence diverges or converges to negative infinity. Sequence_sentence_92

## Series Sequence_section_16

Main article: Series (mathematics) Sequence_sentence_93

## Use in other fields of mathematics Sequence_section_17

### Topology Sequence_section_18

Sequences play an important role in topology, especially in the study of metric spaces. Sequence_sentence_94

For instance: Sequence_sentence_95

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• A metric space is compact exactly when it is sequentially compact.Sequence_item_2_5
• A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences.Sequence_item_2_6
• A metric space is a connected space if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.Sequence_item_2_7
• A topological space is separable exactly when there is a dense sequence of points.Sequence_item_2_8

Sequences can be generalized to nets or filters. Sequence_sentence_96

These generalizations allow one to extend some of the above theorems to spaces without metrics. Sequence_sentence_97

#### Product topology Sequence_section_19

The topological product of a sequence of topological spaces is the cartesian product of those spaces, equipped with a natural topology called the product topology. Sequence_sentence_98

### Analysis Sequence_section_20

In analysis, when talking about sequences, one will generally consider sequences of the form Sequence_sentence_99

which is to say, infinite sequences of elements indexed by natural numbers. Sequence_sentence_100

It may be convenient to have the sequence start with an index different from 1 or 0. Sequence_sentence_101

For example, the sequence defined by xn = 1/log(n) would be defined only for n ≥ 2. Sequence_sentence_102

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 N. Sequence_sentence_103

The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. Sequence_sentence_104

This type can be generalized to sequences of elements of some vector space. Sequence_sentence_105

In analysis, the vector spaces considered are often function spaces. Sequence_sentence_106

Even more generally, one can study sequences with elements in some topological space. Sequence_sentence_107

#### Sequence spaces Sequence_section_21

Main article: Sequence space Sequence_sentence_108

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Sequence_sentence_109

Equivalently, it is a function space whose elements are functions from the natural numbers to the field K, where K is either the field of real numbers or the field of complex numbers. Sequence_sentence_110

The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. Sequence_sentence_111

All sequence spaces are linear subspaces of this space. Sequence_sentence_112

Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. Sequence_sentence_113

The most important sequences spaces in analysis are the ℓ spaces, consisting of the p-power summable sequences, with the p-norm. Sequence_sentence_114

These are special cases of L spaces for the counting measure on the set of natural numbers. Sequence_sentence_115

Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Sequence_sentence_116

Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called an FK-space. Sequence_sentence_117

### Linear algebra Sequence_section_22

Sequences over a field may also be viewed as vectors in a vector space. Sequence_sentence_118

Specifically, the set of F-valued sequences (where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers. Sequence_sentence_119

### Abstract algebra Sequence_section_23

Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or rings. Sequence_sentence_120

#### Free monoid Sequence_section_24

Main article: Free monoid Sequence_sentence_121

If A is a set, the free monoid over A (denoted A, also called Kleene star of A) is a monoid containing all the finite sequences (or strings) of zero or more elements of A, with the binary operation of concatenation. Sequence_sentence_122

The free semigroup A is the subsemigroup of A containing all elements except the empty sequence. Sequence_sentence_123

#### Exact sequences Sequence_section_25

Main article: Exact sequence Sequence_sentence_124

In the context of group theory, a sequence Sequence_sentence_125

of groups and group homomorphisms is called exact, if the image (or range) of each homomorphism is equal to the kernel of the next: Sequence_sentence_126

The sequence of groups and homomorphisms may be either finite or infinite. Sequence_sentence_127

A similar definition can be made for certain other algebraic structures. Sequence_sentence_128

For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms. Sequence_sentence_129

#### Spectral sequences Sequence_section_26

Main article: Spectral sequence Sequence_sentence_130

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Sequence_sentence_131

Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (), they have become an important research tool, particularly in homotopy theory. Sequence_sentence_132

### Set theory Sequence_section_27

An ordinal-indexed sequence is a generalization of a sequence. Sequence_sentence_133

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. Sequence_sentence_134

In this terminology an ω-indexed sequence is an ordinary sequence. Sequence_sentence_135

### Computing Sequence_section_28

In computer science, finite sequences are called lists. Sequence_sentence_136

Potentially infinite sequences are called streams. Sequence_sentence_137

Finite sequences of characters or digits are called strings. Sequence_sentence_138

### Streams Sequence_section_29

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. Sequence_sentence_139

They are often referred to simply as sequences or streams, as opposed to finite strings. Sequence_sentence_140

Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0, 1}). Sequence_sentence_141

The set C = {0, 1} of all infinite binary sequences is sometimes called the Cantor space. Sequence_sentence_142

An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to 1 if and only if the n th string (in shortlex order) is in the language. Sequence_sentence_143

This representation is useful in the diagonalization method for proofs. Sequence_sentence_144

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