TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a sequence is an enumerated collection of objects in which repetitions are allowed and
order Order or ORDER or Orders may refer to: * Orderliness Orderliness is associated with other qualities such as cleanliness Cleanliness is both the abstract state of being clean and free from germs, dirt, trash, or waste, and the habit of achieving a ...
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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, defined as a function from an index set that may not be numbers to another set of elements. 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 Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
'', as in these examples, or ''
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...
'', such as the sequence of all even
positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, 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, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F_0=0,\quad F_1= 1, and F_n= ...

''$F$'' is generally denoted as ''$F_n$''. In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and software. It has sci ...

and
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, finite sequences are sometimes called
strings String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
,
words In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis inclu ...
or
lists A ''list'' is any set of items. List or lists may also refer to: People * List (surname)List or Liste is a European surname. Notable people with the surname include: List * Friedrich List (1789–1846), German economist * Garrett List (1943 ...
, the different names commonly corresponding to different ways to represent them in
computer memory In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
; infinite sequences are called
streamsIn computer networking, STREAMS is the native framework in UNIX System V, Unix System V for implementing character device drivers, network protocols, and inter-process communication. In this framework, a stream is a chain of coroutines that message p ...
. 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, spaces, and other mathematical structures using the
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
properties of sequences. In particular, sequences are the basis for
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
, which are important in
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
and
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
. 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 (mathematics), 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 ...
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 , , or (as a single glyph) , also known informally as dot-dot-dot, is a series of (usually three) dots that indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning. ...

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 (mathematics), 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 ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

greater than 1 that have no
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, particularly in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

where many results related to them exist. The
Fibonacci numbers In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F_0=0,\quad F_1= 1, and F_n=F_ + F_ for . Th ...

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
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. 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 a researcher at AT&T Labs. He transferred the intellectual property Intellectual property (I ...
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 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 megastructure in the ''Halo'' series ...
, and the set of values that it can take is called the
index set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

or
infimum 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 all elements of S, if such an element exists. Consequently, the term ''greatest low ...
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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

. 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 Linguistics is the scientific study of language, meaning tha ...
. 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, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F_0=0,\quad F_1= 1, and F_n= ...

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 sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it cont ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s in . For most holonomic sequences, there is no explicit formula for expressing explicitly $a_n$ as a function of . Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many
special functions Special functions are particular mathematical function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), ...
have a
Taylor series In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...
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 (mathematics), 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 ...
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 An exact sequence is a sequence of morphisms between objects (for example, groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same ...
) are not covered by the definitions and notations introduced below.

## Definition

function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
is an interval of
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

. 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. 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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

)
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 (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, and −2048 are integers, while 9 ...
is a sequence whose terms are integers. * A
polynomial sequence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
. 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 A bitstream (or bit stream), also known as binary sequence, is a sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it conta ...
is a sequence whose terms have one of two discrete values, e.g.
base 2 Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium's three mobile telephone company, telecommunications operators. It is a subsidiary of Telenet (Belgium), ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, 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 measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...
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 In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
)
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 Figure 3. A function that is ''not'' monotonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geometr ...
, and, in particular, in
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies Here Technologies (trading as A trade name, trading name, or business name is a pseudonym A pseudonym () or alias () (originally: ...
. More generally, any sequence of rational numbers that converges to an
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geometr ...
. For instance: * A
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
exactly when it is
sequentially compactIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. * A function from a metric space to another metric space is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
exactly when it takes convergent sequences to convergent sequences. * A metric space is a
connected space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is separable exactly when there is a dense sequence of points. Sequences can be generalized to nets or
filters Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
. These generalizations allow one to extend some of the above theorems to spaces without metrics.

### Product topology

The
topological product In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
of a sequence of topological spaces is the
cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 t ...
called the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
. 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 topologyIn topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the c ...
(i.e. the topology with the fewest open sets) for which all the projections ''pi'' are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
. The product topology is sometimes called the Tychonoff topology.

## Analysis

In
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
, 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 File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s. It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by ''xn'' = 1/
log Log most often refers to: * Trunk (botany) In botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who specialises in thi ...

(''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 Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

numbers. This type can be generalized to sequences of elements of some
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. In analysis, the vector spaces considered are often
function space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s. Even more generally, one can study sequences with elements in some
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
.

### Sequence spaces

A sequence space is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
whose elements are infinite sequences of
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

numbers. Equivalently, it is a
function space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

to the Field (mathematics), 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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm (mathematics), norm, or at least the structure of a topological vector space. 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 Lp space, L''p'' spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or Sequence_space#c,_c0_and_c00, 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 of pointwise convergence, under which it becomes a special kind of Fréchet space called an FK-space.

## Linear algebra

Sequences over a field (mathematics), field may also be viewed as Vector (geometric), vectors in a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. Specifically, the set of ''F''-valued sequences (where ''F'' is a field) is a
function space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
(in fact, a product space) 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 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. The free semigroup ''A''+ is the subsemigroup of ''A''* containing all elements except the empty sequence.

### Exact sequences

In the context of group theory, a sequence :$G_0 \;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n$ of group (mathematics), groups and group homomorphisms is called exact, if the Image (mathematics), image (or Range of a function, range) of each homomorphism is equal to the Kernel (algebra), kernel 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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 sequence An exact sequence is a sequence of morphisms between objects (for example, groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same ...
s, 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 deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, finite sequences are called
lists A ''list'' is any set of items. List or lists may also refer to: People * List (surname)List or Liste is a European surname. Notable people with the surname include: List * Friedrich List (1789–1846), German economist * Garrett List (1943 ...
. 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 Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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 a researcher at AT&T Labs. He transferred the intellectual property Intellectual property (I ...
* 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