TheInfoList

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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, it contains members (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, 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 ...
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, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in ...
, 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 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 the sequence from ...
''$F$'' 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 hardware and software. Computing has scientific, e ...
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 practical disciplines (includi ...
, finite sequences are sometimes called strings,
words A word is a basic element of language that carries an objective or practical meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no cons ...
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 electronic devices. The term ''memory'' is often synonymous with the term '' primary storage ...
; 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), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four We ...
properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
. 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 ways ...
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 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 ways ...
s are the natural numbers greater than 1 that have no divisors 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 integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
where many results related to them exist. The
Fibonacci numbers In mathematics, the Fibonacci numbers, commonly denoted , form a 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 the sequence from ...
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,
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 ...
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, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consist ...
. 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 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 numbers 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 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. 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 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 the sequence from ...
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. 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 consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
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 Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined ...
have a
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
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 ways ...
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. 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. 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 is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
. 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, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
) directed set 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 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 between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
. 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 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 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, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
. 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 together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, 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 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, and, in particular, in real analysis. 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 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, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
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 spaces 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. For instance: * A
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
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 and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
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 Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component t ...
. 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 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 notation, that is : A\tim ...
of those spaces, equipped with a natural topology called the product topology. 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 ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
, 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 Log most often refers to: * Trunk (botany), the stem and main wooden axis of a tree, called logs when cut ** Logging, cutting down trees for logs ** Firewood, logs used for fuel ** Lumber or timber, converted from wood logs * Logarithm, in mathem ...
(''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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers. This type can be generalized to sequences of elements of some
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
. In analysis, the vector spaces considered are often
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s. Even more generally, one can study sequences with elements in some
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
.

### Sequence spaces

A sequence space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
whose elements are infinite sequences of real or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers. Equivalently, it is a
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
whose elements are functions from the natural numbers to the field ''K'', where ''K'' is either the field of real numbers or the field of complex numbers. 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 and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 flats and affine subspaces. In the case of vector spaces over the reals, ...
s of this space. Sequence spaces are typically equipped with a
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
, or at least the structure of a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. 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 on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...
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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of
pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
, 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 and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
. Specifically, the set of ''F''-valued sequences (where ''F'' is a field) is a
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
(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 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 groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, a sequence :$G_0 \;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n$ of groups and group homomorphisms 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-dimension ...
(or range) of each homomorphism is equal to the 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 structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s. For example, one could have an exact sequence of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s and
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s, or of modules and module homomorphisms.

### Spectral sequences

In homological algebra and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
, 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 In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
.

## Set theory

An 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 practical disciplines (includi ...
, finite sequences are called lists. Potentially 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 ...
. Finite sequences of characters or digits are called
string 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), a Canadian ani ...
s.

## Streams

Infinite sequences of digits (or characters) drawn from a finite
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a sylla ...
are of particular interest in
theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the t ...
. They are often referred to simply as ''sequences'' or ''
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 ...
'', as opposed to finite '' 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 In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sym ...
(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 diagonalization method for proofs.

*
Enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (fo ...
*
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 ...
* Recurrence relation * Sequence space ;Operations * Cauchy product ;Examples *
Discrete-time signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
*
Farey sequence In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in orde ...
*
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a 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 the sequence from ...
* Look-and-say sequence * Thue–Morse sequence * List of integer sequences ;Types * ±1-sequence *
Arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
* Automatic sequence *
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
*
Constant-recursive sequence In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. A constan ...
* Geometric progression * Harmonic progression * Holonomic sequence * Regular sequence * Pseudorandom binary sequence *
Random sequence The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let ''X''1,...,''Xn'' be independ ...
;Related concepts *
List (computing) In computer science, a list or sequence is an abstract data type that represents a finite number of ordered values, where the same value may occur more than once. An instance of a list is a computer representation of the mathematical concept of ...
* Net (topology) (a generalization of sequences) * Ordinal-indexed sequence *
Recursion (computer science) In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves ...
*
Set (mathematics) A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or ...
*
Tuple In mathematics, a tuple is a finite ordered list ( sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is de ...
*
Permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...