HOME

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 Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
in which repetitions are allowed and
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
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 ...
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 Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
positive integers (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 ...
, 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 ''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, ...
and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, finite sequences are sometimes called strings, words or
lists A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ...
, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. 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 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 (3 ...
. 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 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 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 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 where many results related to them exist. The Fibonacci numbers 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. 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'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of a sequence of rational numbers (e.g. via its
decimal expansion A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, ...
). 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 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 (2n)_. The sequence of squares could be written as (n^2)_. The variable ''n'' is called an index, and the set of values that it can take is called the index set. 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 (a_n)_, which denotes a sequence whose ''n''th element is given by the variable a_n. For example: :\begin a_1 &= 1\text(a_n)_ \\ a_2 &= 2\text \\ a_3 &= 3\text \\ &\;\;\vdots \\ a_ &= (n-1)\text \\ a_n &= n\text \\ a_ &= (n+1)\text \\ &\;\; \vdots \end One can consider multiple sequences at the same time by using different variables; e.g. (b_n)_ could be a different sequence than (a_n)_. One can even consider a sequence of sequences: ((a_)_)_ denotes a sequence whose ''m''th term is the sequence (a_)_. 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 (k^2)_^ denotes the ten-term sequence of squares (1, 4, 9, \ldots, 100). The limits \infty and -\infty are allowed, but they do not represent valid values for the index, only the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
or infimum of such values, respectively. For example, the sequence (a_n)_^\infty is the same as the sequence (a_n)_, and does not contain an additional term "at infinity". The sequence (a_n)_^\infty is a bi-infinite sequence, and can also be written as (\ldots, a_, a_0, a_1, a_2, \ldots). In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes (a_k) 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 :(a_k)_^\infty = ( a_0, a_1, a_2, \ldots ). 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. * (1, 9, 25, \ldots) * (a_1, a_3, a_5, \ldots), \qquad a_k = k^2 * (a_)_^\infty, \qquad a_k = k^2 * (a_)_^\infty, \qquad a_k = (2k-1)^2 * \left((2k-1)^2\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 (a_)_^\infty, but it is not the same as the sequence denoted by the expression.


Defining a sequence by recursion

Sequences whose elements are related to the previous elements in a straightforward way are often defined using
recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...
. 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 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 Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...
. 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 polynomials in . For most holonomic sequences, there is no explicit formula for expressing a_n as a function of . Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series 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) are not covered by the definitions and notations introduced below.


Definition

In this article, a sequence is formally defined as a function 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 * ...
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 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 poin ...
. 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 (a_n)_, or just as (a_n). 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)
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
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 (2n)_^.


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 (a_n)_^ 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. 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 (a_n)_ is any sequence of the form (a_)_, where (n_k)_ 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 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 = (-1)^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 (a_n) is normally denoted \lim_a_n. If (a_n) is a divergent sequence, then the expression \lim_a_n is meaningless.


Formal definition of convergence

A sequence of real numbers (a_n) 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 (a_n) 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 (a_n) 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(a_n, L), which denotes the distance between a_n and L.


Applications and important results

If (a_n) and (b_n) are convergent sequences, then the following limits exist, and can be computed as follows: * \lim_ (a_n \pm b_n) = \lim_ a_n \pm \lim_ b_n * \lim_ c a_n = c \lim_ a_n for all real numbers c * \lim_ (a_n b_n) = \left( \lim_ a_n \right) \left( \lim_ b_n \right) * \lim_ \frac = \frac, provided that \lim_ b_n \ne 0 * \lim_ a_n^p = \left( \lim_ a_n \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 (c_n) is a sequence such that a_n \leq c_n \leq b_n for all n > N
then (c_n) 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 Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
. More generally, any sequence of rational numbers that converges to an irrational number is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers. 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 (a_n) 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 (S_N)_, 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 Britis ...
exactly when it is sequentially compact. * 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 g ...
exactly when it takes convergent sequences to convergent sequences. * A metric space is a connected space if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set. * 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 poin ...
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 that ...
. These generalizations allow one to extend some of the above theorems to spaces without metrics.


Product topology

The
topological product In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
of a sequence of topological spaces is the cartesian product of those spaces, equipped with a natural topology called the product topology. More formally, given a sequence of spaces (X_i)_, the product space :X := \prod_ X_i, is defined as the set of all sequences (x_i)_ 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((x_j)_) = 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 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 g ...
. 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 (3 ...
, when talking about sequences, one will generally consider sequences of the form :(x_1, x_2, x_3, \dots)\text(x_0, x_1, x_2, \dots) 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 ...
s. A sequence may start with an index different from 1 or 0. For example, the sequence defined by ''xn'' = 1/ log(''n'') would be defined only for ''n'' ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given ''N''. The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. 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 poin ...
.


Sequence spaces

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field ''K'', where ''K'' is either the field of real numbers or the field of complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in ''K'', and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a 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 L''p'' spaces for the counting measure 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 an ...
, 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. Specifically, the set of ''F''-valued sequences (where ''F'' is a field) is a function space (in fact, a product space) of ''F''-valued functions over the set of natural numbers.


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. 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 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 ...
, a sequence :G_0 \;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
and group homomorphisms is called exact, if the image (or range) of each homomorphism is equal to the kernel of the next: :\mathrm(f_k) = \mathrm(f_) 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 spaces and linear maps, or of
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
and module homomorphisms.


Spectral sequences

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


Set theory

An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and ''X'' is a set, an α-indexed sequence of elements of ''X'' is a function from α to ''X''. In this terminology an ω-indexed sequence is an ordinary sequence.


Computing

In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, finite sequences are called
lists A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ...
. Potentially infinite sequences are called streams. Finite sequences of characters or digits are called strings.


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 syllab ...
are of particular interest in theoretical computer science. They are often referred to simply as ''sequences'' or '' streams'', 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.


See also

* Enumeration * On-Line Encyclopedia of Integer Sequences * Recurrence relation * Sequence space ;Operations * Cauchy product ;Examples * Discrete-time signal * Farey sequence * 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 * Holonomic sequence * Regular sequence * Pseudorandom binary sequence * Random sequence ;Related concepts * List (computing) * Net (topology) (a generalization of sequences) * Ordinal-indexed sequence * Recursion (computer science) *
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, o ...
*
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 defi ...
* Permutation


Notes


References


External links

*
The On-Line Encyclopedia of Integer Sequences


(free) {{Authority control Elementary mathematics *