In mathematics, a **sequence** is an ordered list of objects (or events). Like a set, it contains members (also called *elements* or *terms*), and the number of terms (possibly infinite) is called the *length* of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence. Sequence can refer to: sequence, a logical and mathematical notion In biochemistry, a biopolymers sequence is synonymous with its primary structure: the list of basic building blocks constituting the polymer (for example a DNA sequence). ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
A Military jury (or Members, in military parlance) serves a function similar to a civilian jury, but with several notable differences. ...
For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be *finite*, as in this example, or *infinite*, such as the sequence of all even positive integers (2, 4, 6,...). In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
The infinity symbol âˆž in several typefaces. ...
In mathematics, the parity of an object refers to whether it is even or odd. ...
A negative number is a number that is less than zero, such as âˆ’3. ...
The integers are commonly denoted by the above symbol. ...
An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent. It is however bounded. Image File history File links Size of this preview: 800 Ã— 512 pixelsFull resolution (2706 Ã— 1733 pixel, file size: 80 KB, MIME type: image/png) % draw an illustration of a sequence that is not Cauchy function main() % prepare the screen and define some parameters figure(1); clf; hold on; axis equal...
Image File history File links Size of this preview: 800 Ã— 512 pixelsFull resolution (2706 Ã— 1733 pixel, file size: 80 KB, MIME type: image/png) % draw an illustration of a sequence that is not Cauchy function main() % prepare the screen and define some parameters figure(1); clf; hold on; axis equal...
## Examples and notation
There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below. E.G. is an Australian only release EP from New Zealand four piece Goodshirt. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
A sequence may be denoted (*a*_{1}, *a*_{2}, ...). For shortness, the notation (*a*_{n}) is also used. A more formal definition of a **finite sequence** with terms in a set *S* is a function from {1, 2, ..., *n*} to *S* for some *n* ≥ 0. An **infinite sequence** in *S* is a function from {1, 2, ...} (the set of natural numbers without 0) to *S*. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
Sequences may also start from 0, so the first term in the sequence is then *a*_{0}. A finite sequence is also called an n-tuple. Finite sequences include the *empty sequence* ( ) that has no elements. In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...
A function from all integers into a set is sometimes called a **bi-infinite sequence**, since it may be thought of as a sequence indexed by negative integers grafted onto a sequence indexed by positive integers.
## Types and properties of sequences 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. In mathematics, a subsequence of some sequence is a new sequence which is formed from the original sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. ...
If the terms of the sequence are a subset of an ordered set, then a *monotonically increasing* sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called *strictly monotonically increasing*. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or *monotone*. This is a special case of the more general notion of monotonic function. In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ...
In mathematical writing, the adjective strict is used in to modify technical terms which have multiple meanings. ...
A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ...
A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ...
The terms *non-decreasing* and *non-increasing* are used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively. If the terms of a sequence are integers, then the sequence is an integer sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence. The integers are commonly denoted by the above symbol. ...
In mathematics, an integer sequence is a sequence (i. ...
In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ...
If *S* is endowed with a topology, then it becomes possible to consider *convergence* of an infinite sequence in *S*. Such considerations involve the concept of the limit of a sequence. A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
The limit of a sequence is one of the oldest concepts in mathematical analysis. ...
## Sequences in analysis In analysis, when talking about sequences, one will generally consider sequences of the form Analysis has its beginnings in the rigorous formulation of calculus. ...
- or
which is to say, infinite sequences of elements indexed by natural numbers. (It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by *x*_{n} = 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*.) In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...
In mathematics, a phrase like ... for all n large enough, means there exists M such that . ...
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 complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
## Series -
The sum of terms of a sequence is a series. More precisely, if (*x*_{1}, *x*_{2}, *x*_{3}, ...) is a sequence, one may consider the sequence of partial sums (*S*_{1}, *S*_{2}, *S*_{3}, ...), with In mathematics, a series is often represented as the sum of a sequence of terms. ...
In mathematics, a series is often represented as the sum of a sequence of terms. ...
In mathematics, a series is a sum of a sequence of terms. ...
Formally, this pair of sequences comprises the *series* with the terms *x*_{1}, *x*_{2}, *x*_{3}, ..., which is denoted as If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see series. In mathematics, a series is often represented as the sum of a sequence of terms. ...
## Infinite sequences in theoretical computer science Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as *sequences* (as opposed to finite *strings*). Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0,1}). The set *C* = {0, 1}^{∞} of all infinite, binary sequences is sometimes called the Cantor space. In mathematics and computer science, a numerical digit is a symbol, e. ...
In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
In computer science, an alphabet is a finite set of characters or digits. ...
Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ...
In computer programming and formal language theory, (and other branches of mathematics), a string is an ordered sequence of symbols. ...
This article is about the unit of information. ...
In mathematics, the term Cantor space is sometimes used to denote the topological abstraction of the classical Cantor set: A topological space is a Cantor space if it is homeomorphic to the Cantor set. ...
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. Therefore, the study of complexity classes, which are sets of languages, may be regarded as studying sets of infinite sequences. In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ...
The shortlex (or radix, or length-plus-lexicographic) order is an ordering for ordered sets of objects, where the sequences are primarily sorted by cardinality (length) with the shortest sequences first, and sequences of the same length are sorted into lexicographical order. ...
In computational complexity theory, a complexity class is a set of problems of related complexity. ...
An infinite sequence drawn from the alphabet {0, 1, ..., b−1} may also represent a real number expressed in the base-*b* positional number system. This equivalence is often used to bring the techniques of real analysis to bear on complexity classes. ...
Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
## Sequences as vectors 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. A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...
In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...
In particular, the term *sequence space* usually refers to a linear subspace of the set of all possible infinite sequences with elements in . In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ...
The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...
## Doubly-infinite sequences Normally, the term *infinite sequence* refers to a sequence which is infinite in one direction, and finite in the other -- the sequence has a first element, but no final element (a *singly-infinite sequence*). A *doubly-infinite sequence* is infinite in both directions -- it has neither a first nor a final element. Singly-infinite sequences are functions from the natural numbers (**N'**) to some set, whereas doubly-infinite sequences are functions from the integers (**Z**) to some set. One can interpret singly infinite sequences as element of the semigroup ring of the natural numbers , and doubly infinite sequences as elements of the group ring of the integers . This perspective is used in the Cauchy product of sequences. In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...
The integers are commonly denoted by the above symbol. ...
In mathematics, the Cauchy product, named in honor of Augustin Louis Cauchy, of two strictly formal (not necessarily convergent) series usually, of real or complex numbers, is defined by a discrete convolution as follows. ...
## Ordinal-indexed sequence An ordinal-indexed sequencehttp://en.wikipedia.org/wiki/Order_topology#Ordinal-indexed_sequences 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. In mathematics, the order topology is a topology that can be defined on any totally ordered set. ...
A limit ordinal is an ordinal number which is not a successor ordinal. ...
## Sequences and automata Automata or finite state machines can typically thought of as directed graphs, with edges labeled using some specific alphabet Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence called a *word* (or input word). The sequence of states encountered by the automaton when processing a word is called a *run*. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter. In theoretical computer science, automata theory is the study of abstract machines and problems they are able to solve. ...
Fig. ...
## See also In mathematics the term net has at least two meanings. ...
In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ...
Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...
In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...
### Types of sequences In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...
In mathematics, an integer sequence is a sequence (i. ...
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. ...
In mathematics and its applications, the Thue-Morse sequence, or Prouhet-Thue-Morse sequence, is a certain binary sequence whose initial segments alternate (in a certain sense). ...
In mathematics, the look-and-say sequence is the sequence of integers beginning as follows: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of...
A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above â€“ see golden spiral. ...
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...
Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ...
### Related concepts Look up list in Wiktionary, the free dictionary. ...
In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...
In mathematics, the order topology is a topology that can be defined on any totally ordered set. ...
### Operations on sequences The limit of a sequence is one of the oldest concepts in mathematical analysis. ...
In mathematics, the Cauchy product, named in honor of Augustin Louis Cauchy, of two strictly formal (not necessarily convergent) series usually, of real or complex numbers, is defined by a discrete convolution as follows. ...
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