sequence

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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains
members Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in ...
(also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an
indexed family In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' infinite'', such as the sequence of all even
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s (2, 4, 6, ...). The position of an element in a sequence is its ''rank'' or ''index''; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a sequence is often denoted by letters in the form of $a_n$, $b_n$ and $c_n$, where the subscript ''n'' refers to the ''n''th element of the sequence; for example, the ''n''th element of the
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
''$F$'' is generally denoted as ''$F_n$''. In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , hardware and software. ...
and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, finite sequences are sometimes called strings,
words A word is a basic element of language that carries an semantics, objective or pragmatics, practical semantics, meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of w ...
or lists, the different names commonly corresponding to different ways to represent them in
computer memory In computing, memory is a device or system that is used to store information for immediate use in a computer or related computer hardware and Digital data, digital Electronics, electronic devices. The term ''memory'' is often synonymous with th ...
; infinite sequences are called
streams A stream is a continuous body of water, body of surface water Current (stream), flowing within the stream bed, bed and bank (geography), banks of a channel (geography), channel. Depending on its location or certain characteristics, a stream ...
. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

# Examples and notation

A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
properties of sequences. In particular, sequences are the basis for series, which are important in
differential equations In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
and
analysis Analysis (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
. Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
s. There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with
ellipsis The ellipsis (, also known informally as dot dot dot) is a series of dots that indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning. The plural is ellipses. The term origin ...
leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting a sequence are discussed after the examples.

## Examples

The
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
s are the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
greater than 1 that have no
divisor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
where many results related to them exist. The
Fibonacci numbers In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
comprise the integer sequence whose elements are the sum of the previous two elements. 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, ...). Other examples of sequences include those made up of
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
,
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s and
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as the limit of a sequence of rational numbers (e.g. via its decimal expansion). As another example, is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of , that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that is easily discernible by inspection. Another example of sequences is a sequence of functions, where each member of the sequence is a function whose shape is determined by a natural number indexing that function. The
On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
comprises a large list of examples of integer sequences.

## Indexing

Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of . One such notation is to write down a general formula for computing the ''n''th term as a function of ''n'', enclose it in parentheses, and include a subscript indicating the set of values that ''n'' can take. For example, in this notation the sequence of even numbers could be written as $\left(2n\right)_$. The sequence of squares could be written as $\left(n^2\right)_$. The variable ''n'' is called an
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
, and the set of values that it can take is called the
index set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
. It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like $\left(a_n\right)_$, which denotes a sequence whose ''n''th element is given by the variable $a_n$. For example: :$\begin a_1 &= 1\text\left(a_n\right)_ \\ a_2 &= 2\text \\ a_3 &= 3\text \\ &\;\;\vdots \\ a_ &= \left(n-1\right)\text \\ a_n &= n\text \\ a_ &= \left(n+1\right)\text \\ &\;\; \vdots \end$ One can consider multiple sequences at the same time by using different variables; e.g. $\left(b_n\right)_$ could be a different sequence than $\left(a_n\right)_$. One can even consider a sequence of sequences: $\left(\left(a_\right)_\right)_$ denotes a sequence whose ''m''th term is the sequence $\left(a_\right)_$. An alternative to writing the domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation $\left(k^2\right)_^$ denotes the ten-term sequence of squares $\left(1, 4, 9, \ldots, 100\right)$. The limits $\infty$ and $-\infty$ are allowed, but they do not represent valid values for the index, only the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
or infimum of such values, respectively. For example, the sequence $\left(a_n\right)_^\infty$ is the same as the sequence $\left(a_n\right)_$, and does not contain an additional term "at infinity". The sequence $\left(a_n\right)_^\infty$ is a bi-infinite sequence, and can also be written as $\left(\ldots, a_, a_0, a_1, a_2, \ldots\right)$. In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes $\left(a_k\right)$ for an arbitrary sequence. Often, the index ''k'' is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in :$\left(a_k\right)_^\infty = \left( a_0, a_1, a_2, \ldots \right).$ In some cases, the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of
odd number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s could be denoted in any of the following ways. * $\left(1, 9, 25, \ldots\right)$ * $\left(a_1, a_3, a_5, \ldots\right), \qquad a_k = k^2$ * $\left(a_\right)_^\infty, \qquad a_k = k^2$ * $\left(a_\right)_^\infty, \qquad a_k = \left(2k-1\right)^2$ * $\left\left(\left(2k-1\right)^2\right\right)_^\infty$ Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
. In the second and third bullets, there is a well-defined sequence $\left(a_\right)_^\infty$, but it is not the same as the sequence denoted by the expression.

## Defining a sequence by recursion

Sequences whose elements are related to the previous elements in a straightforward way are often defined using
recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
. This is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one needs a rule, called ''recurrence relation'' to construct each element in terms of the ones before it. 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. The
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
is a simple classical example, defined by the recurrence relation :$a_n = a_ + a_,$ with initial terms $a_0 = 0$ and $a_1 = 1$. From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. A complicated example of a sequence defined by a recurrence relation is Recamán's sequence, defined by the recurrence relation :$\begina_n = a_ - n,\quad \text\\a_n = a_ + n, \quad\text, \end$ with initial term $a_0 = 0.$ A ''linear recurrence with constant coefficients'' is a recurrence relation of the form :$a_n=c_0 +c_1a_+\dots+c_k a_,$ where $c_0,\dots, c_k$ are constants. There is a general method for expressing the general term $a_n$ of such a sequence as a function of ; see Linear recurrence. In the case of the Fibonacci sequence, one has $c_0=0, c_1=c_2=1,$ and the resulting function of is given by
Binet's formula In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
. A holonomic sequence is a sequence defined by a recurrence relation of the form :$a_n=c_1a_+\dots+c_k a_,$ where $c_1,\dots, c_k$ are
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s in . For most holonomic sequences, there is no explicit formula for expressing $a_n$ as a function of . Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a
Taylor series In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions. Not all sequences can be specified by a recurrence relation. An example is the sequence of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
s in their natural order (2, 3, 5, 7, 11, 13, 17, ...).

# Formal definition and basic properties

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.

## Definition

In this article, a sequence is formally defined as a function whose domain is an interval of
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
. 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). However, many authors use a narrower definition by requiring the domain of a sequence to be the set of
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
. 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. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition, the
codomain In mathematics, the codomain or set of destination of a Function (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...
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 In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, rather than . There are terminological differences as well: the value of a sequence at the lowest input (often 1) is called the "first element" of the sequence, the value at the second smallest input (often 2) is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. ''f'', a sequence abstracted from its input is usually written by a notation such as $\left(a_n\right)_$, or just as $\left(a_n\right).$ Here is the domain, or index set, of the sequence. Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets. A net is a function from a (possibly
uncountable In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
)
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
to a topological space. The notational conventions for sequences normally apply to nets as well.

## Finite and infinite

The length of a sequence is defined as the number of terms in the sequence. A sequence of a finite length ''n'' is also called an ''n''-tuple. Finite sequences include the empty sequence ( ) that has no elements. Normally, the term ''infinite sequence'' refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence is called a singly infinite sequence or a one-sided infinite sequence when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence. A function from the set Z of ''all''
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ), is bi-infinite. This sequence could be denoted $\left(2n\right)_^$.

## Increasing and decreasing

A sequence is said to be ''monotonically increasing'' if each term is greater than or equal to the one before it. For example, the sequence $\left(a_n\right)_^$ is monotonically increasing if and only if ''a''''n''+1 $\geq$ ''a''''n'' for all ''n'' ∈ N. If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonically decreasing if each consecutive term is less than or equal to the previous one, and is strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a
monotonic function In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...
. 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.

## Bounded

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. In other words, this means that there exists ''M'' such that for all ''n'', ''an'' ≤ ''M''. Any such ''M'' is called an ''upper bound''. 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''. If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded.

## Subsequences

A
subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
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. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of the positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved. Formally, a subsequence of the sequence $\left(a_n\right)_$ is any sequence of the form $\left(a_\right)_$, where $\left(n_k\right)_$ is a strictly increasing sequence of positive integers.

## Other types of sequences

Some other types of sequences that are easy to define include: * An integer sequence is a sequence whose terms are integers. * A
polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each indexed family, index is equal to the degree of a polynomial, degree of the corresponding polynomial. Polynomial ...
is a sequence whose terms are polynomials. * A positive integer sequence is sometimes called multiplicative, if ''a''''nm'' = ''a''''n'' ''a''''m'' for all pairs ''n'', ''m'' such that ''n'' and ''m'' are
coprime In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
. In other instances, sequences are often called ''multiplicative'', if ''a''''n'' = ''na''1 for all ''n''. Moreover, a ''multiplicative'' Fibonacci sequence satisfies the recursion relation ''a''''n'' = ''a''''n''−1 ''a''''n''−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.

# Limits and convergence

An important property of a sequence is ''convergence''. If a sequence converges, it converges to a particular value known as the ''limit''. If a sequence converges to some limit, then it is convergent. A sequence that does not converge is divergent. Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value $L$ (called the limit of the sequence), and they become and remain ''arbitrarily'' close to $L$, meaning that given a real number $d$ greater than zero, all but a finite number of the elements of the sequence have a distance from $L$ less than $d$. For example, the sequence $a_n = \frac$ shown to the right converges to the value 0. On the other hand, the sequences $b_n = n^3$ (which begins 1, 8, 27, …) and $c_n = \left(-1\right)^n$ (which begins −1, 1, −1, 1, …) are both divergent. If a sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence $\left(a_n\right)$ is normally denoted $\lim_a_n$. If $\left(a_n\right)$ is a divergent sequence, then the expression $\lim_a_n$ is meaningless.

## Formal definition of convergence

A sequence of real numbers $\left(a_n\right)$ converges to a real number $L$ if, for all $\varepsilon > 0$, there exists a natural number $N$ such that for all $n \geq N$ we have :$, a_n - L, < \varepsilon.$ If $\left(a_n\right)$ is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that $, \cdot,$ denotes the complex modulus, i.e. $, z, = \sqrt$. If $\left(a_n\right)$ is a sequence of points in a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
, then the formula can be used to define convergence, if the expression $, a_n-L,$ is replaced by the expression $\operatorname\left(a_n, L\right)$, which denotes the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between $a_n$ and $L$.

## Applications and important results

If $\left(a_n\right)$ and $\left(b_n\right)$ are convergent sequences, then the following limits exist, and can be computed as follows: * $\lim_ \left(a_n \pm b_n\right) = \lim_ a_n \pm \lim_ b_n$ * $\lim_ c a_n = c \lim_ a_n$ for all real numbers $c$ * $\lim_ \left(a_n b_n\right) = \left\left( \lim_ a_n \right\right) \left\left( \lim_ b_n \right\right)$ * $\lim_ \frac = \frac$, provided that $\lim_ b_n \ne 0$ * $\lim_ a_n^p = \left\left( \lim_ a_n \right\right)^p$ for all $p > 0$ and $a_n > 0$ Moreover: * If $a_n \leq b_n$ for all $n$ greater than some $N$, then $\lim_ a_n \leq \lim_ b_n$. * ( Squeeze Theorem)
If $\left(c_n\right)$ is a sequence such that $a_n \leq c_n \leq b_n$ for all $n > N$
then $\left(c_n\right)$ is convergent, and $\lim_ c_n = L$. * If a sequence is bounded and
monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...
then it is convergent. * A sequence is convergent if and only if all of its subsequences are convergent.

## Cauchy sequences

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in
metric spaces In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
, and, in particular, in
real analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
. One particularly important result in real analysis is ''Cauchy characterization of convergence for sequences'': :A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy. In contrast, there are Cauchy sequences of
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
that are not convergent in the rationals, e.g. the sequence defined by ''x''1 = 1 and ''x''''n''+1 = is Cauchy, but has no rational limit, cf. here. More generally, any sequence of rational numbers that converges to an
irrational number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers. Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in . Intuitively, a space is complete if ther ...
s and are particularly nice for analysis.

## Infinite limits

In calculus, it is common to define notation for sequences which do not converge in the sense discussed above, but which instead become and remain arbitrarily large, or become and remain arbitrarily negative. If $a_n$ becomes arbitrarily large as $n \to \infty$, we write :$\lim_a_n = \infty.$ In this case we say that the sequence diverges, or that it converges to infinity. An example of such a sequence is . If $a_n$ becomes arbitrarily negative (i.e. negative and large in magnitude) as $n \to \infty$, we write :$\lim_a_n = -\infty$ and say that the sequence diverges or converges to negative infinity.

# Series

A series is, informally speaking, the sum of the terms of a sequence. That is, it is an expression of the form $\sum_^\infty a_n$ or $a_1 + a_2 + \cdots$, where $\left(a_n\right)$ is a sequence of real or complex numbers. The partial sums of a series are the expressions resulting from replacing the infinity symbol with a finite number, i.e. the ''N''th partial sum of the series $\sum_^\infty a_n$ is the number :$S_N = \sum_^N a_n = a_1 + a_2 + \cdots + a_N.$ The partial sums themselves form a sequence $\left(S_N\right)_$, which is called the sequence of partial sums of the series $\sum_^\infty a_n$. If the sequence of partial sums converges, then we say that the series $\sum_^\infty a_n$ is convergent, and the limit $\lim_ S_N$ is called the value of the series. The same notation is used to denote a series and its value, i.e. we write $\sum_^\infty a_n = \lim_ S_N$.

# Use in other fields of mathematics

## Topology

Sequences play an important role in topology, especially in the study of
metric spaces In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
. For instance: * A
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
is compact exactly when it is sequentially compact. * A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences. * A metric space is a
connected space In topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, de ...
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. * A
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is separable exactly when there is a dense sequence of points. Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems to spaces without metrics.

### Product topology

The topological product of a sequence of topological spaces is the
cartesian product In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...
of those spaces, equipped with a
natural topology In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that ...
called the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. More formally, given a sequence of spaces $\left(X_i\right)_$, the product space :$X := \prod_ X_i,$ is defined as the set of all sequences $\left(x_i\right)_$ such that for each ''i'', $x_i$ is an element of $X_i$. The canonical projections are the maps ''pi'' : ''X'' → ''Xi'' defined by the equation $p_i\left(\left(x_j\right)_\right) = x_i$. Then the product topology on ''X'' is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections ''pi'' are continuous. The product topology is sometimes called the Tychonoff topology.

## Analysis

In
analysis Analysis (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
, when talking about sequences, one will generally consider sequences of the form :$\left(x_1, x_2, x_3, \dots\right)\text\left(x_0, x_1, x_2, \dots\right)$ which is to say, infinite sequences of elements indexed by
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. A sequence may start with an index different from 1 or 0. For example, the sequence defined by ''xn'' = 1/ log(''n'') would be defined only for ''n'' ≥ 2. 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''. 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 mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. In analysis, the vector spaces considered are often
function space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
s. Even more generally, one can study sequences with elements in some
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
.

### Sequence spaces

A
sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real number, real or complex numbers. Equivalently, it is a function space whose elements are functions from the ...
is a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
whose elements are infinite sequences of real or complex numbers. Equivalently, it is a
function space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
whose elements are functions from the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
to the field ''K'', where ''K'' is either the field of real numbers or the field of complex numbers. 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flat (geometry), flats and affine subspaces. In the case of vector spaces o ...
s of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a
topological vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
. The most important sequences spaces in analysis are the ℓ''p'' spaces, consisting of the ''p''-power summable sequences, with the ''p''-norm. These are special cases of L''p'' spaces for the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a Measure (mathematics), measure on any Set (mathematics), set – the "size" of a subset is taken to be the number of elements in the subset if the subset ...
on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted ''c'' and ''c''0, with the sup norm. Any sequence space can also be equipped with the
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
of
pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it i ...
, under which it becomes a special kind of Fréchet space called an FK-space.

## Linear algebra

Sequences over a field may also be viewed as vectors in a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. Specifically, the set of ''F''-valued sequences (where ''F'' is a field) is a
function space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
(in fact, a
product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
) of ''F''-valued functions over the set of natural numbers.

## Abstract algebra

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

### Free monoid

If ''A'' is a set, the
free monoid In abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), f ...
over ''A'' (denoted ''A''*, also called
Kleene star In mathematical logic Mathematical logic is the study of formal logic within mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...
of ''A'') is a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
containing all the finite sequences (or strings) of zero or more elements of ''A'', with the binary operation of concatenation. The free semigroup ''A''+ is the subsemigroup of ''A''* containing all elements except the empty sequence.

### Exact sequences

In the context of
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a sequence :$G_0 \;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n$ of groups and
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
s is called exact, if the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
(or range) of each homomorphism is equal to the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of the next: :$\mathrm\left(f_k\right) = \mathrm\left(f_\right)$ The sequence of groups and homomorphisms may be either finite or infinite. A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s and linear maps, or of module (mathematics), modules and module homomorphisms.

### Spectral sequences

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become an important research tool, particularly in homotopy theory.

## Set theory

An Order topology#Ordinal-indexed sequences, ordinal-indexed sequence is a generalization of a sequence. 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''. In this terminology an ω-indexed sequence is an ordinary sequence.

## Computing

In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, finite sequences are called lists. Potentially infinite sequences are called stream (computer science), streams. Finite sequences of characters or digits are called String (computer science), strings.

## Streams

Infinite sequences of numerical digit, digits (or character (computing), characters) drawn from a finite alphabet (computer science), alphabet are of particular interest in theoretical computer science. They are often referred to simply as ''sequences'' or ''Stream (computing), streams'', as opposed to finite ''String (computer science)#Formal theory, strings''. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet ). The set ''C'' = of all infinite binary sequences is sometimes called the Cantor space. 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. This representation is useful in the Cantor's diagonal argument, diagonalization method for proofs.

* Enumeration *
On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
* Recurrence relation * Sequence space ;Operations * Cauchy product ;Examples * Discrete-time signal * Farey sequence * Fibonacci number, Fibonacci sequence * Look-and-say sequence * Thue–Morse sequence * List of integer sequences ;Types * ±1-sequence * Arithmetic progression * Automatic sequence * Cauchy sequence * Constant-recursive sequence * Geometric progression * Harmonic progression (mathematics), Harmonic progression * holonomic function, Holonomic sequence * k-regular sequence, Regular sequence * Pseudorandom binary sequence * Random sequence ;Related concepts * List (computing) * Net (topology) (a generalization of sequences) * Order topology#Ordinal-indexed sequences, Ordinal-indexed sequence * Recursion (computer science) * Set (mathematics) * Tuple * Permutation

# References

*
The On-Line Encyclopedia of Integer Sequences

(free) {{Authority control Elementary mathematics Sequences and series, *