In mathematics, a sequence is an ordered list of objects (or events). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence.

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. In Set theory, an infinite set is a set that is not a Finite set. In Mathematics, the parity of an object states whether it is even or odd A negative number is a Number that is less than zero, such as −2 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French . . ).

An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent. It is however bounded.

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. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group

A sequence may be denoted (a₁, a₂, . . . ). 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, . The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function . . , 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 natural number (also called counting number) can mean either an element of the set (the positive Integers or an

Sequences may also start from 0, so the first term in the sequence is then a₀.

A sequence of a fixed-length n is also called an n-tuple. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple Finite sequences include the empty sequence ( ) that has no elements.

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

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. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In mathematical writing the adjective strict is used to modify technical terms which have multiple meanings 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.

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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, an integer sequence is a Sequence (ie an ordered list of Integers An integer sequence may be specified explicitly by If the terms of a sequence are polynomials, then the sequence is a polynomial sequence. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a polynomial sequence is a Sequence of Polynomials indexed by the nonnegative integers 0 1 2 3.

If S is endowed with a topology, then it becomes possible to consider convergence of an infinite sequence in S. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Such considerations involve the concept of the limit of a sequence. 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

or

which is to say, infinite sequences of elements indexed by natural numbers. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

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. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce 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, the phrase sufficiently large is used in contexts such as P is true for sufficiently large x which is actually shorthand )

The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This type can be generalized to sequences of elements of some vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In analysis, the vector spaces considered are often function spaces. In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. Even more generally, one can study sequences with elements in some topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

Series

Main article: Series (mathematics)

The sum of terms of a sequence is a series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with More precisely, if (x₁, x₂, x₃, . . . ) is a sequence, one may consider the sequence of partial sums (S₁, S₂, S₃, . In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with . . ), with

Formally, this pair of sequences comprises the series with the terms x₁, x₂, x₃, . . . , 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 That is a series is represented as a list of numbers with

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. In Mathematics and Computer science, a digit is a symbol (a number symbol e For other uses see Character. In Computer and machine-based Telecommunications terminology a character is a unit of In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Computer science, an alphabet is a usually finite set of characters or digits Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such They are often referred to simply as sequences (as opposed to finite strings). In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0,1}). A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication The set C = {0, 1}^∞ of all infinite, binary sequences is sometimes called the Cantor space. In Mathematics, the term Cantor space is sometimes used to denotethe topological abstraction of the classical Cantor set:A Topological space is aCantor

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. A formal language is a set of words, ie finite strings of letters, or symbols. The shortlex (or radix, or length-plus-lexicographic) order is an ordering for Ordered sets of objects where the sequences are primarily sorted by Therefore, the study of complexity classes, which are sets of languages, may be regarded as studying sets of infinite sequences. 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. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a 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 In particular it deals with the analytic properties of real

Sequences as vectors

Sequences over a field may also be viewed as vectors in a vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural

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 a Vector space whose elements are infinite Sequences of Complex 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 . In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This perspective is used in the Cauchy product of sequences. In Mathematics, the Cauchy product, named in honor of Augustin Louis Cauchy, of two Sequences a_n b_n is the discrete Convolution

Ordinal-indexed sequence

An ordinal-indexed sequencehttp://en.wikipedia.org../../../../articles/o/r/d/Order_topology.html#Ordinal-indexed_sequences is a generalization of a sequence. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. 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. A limit ordinal is an Ordinal number which is neither zero nor a Successor ordinal. In this terminology an ω-indexed sequence is an ordinary sequence.

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.

See also

• Net (topology) (a generalization of sequences)
• Sequence space
• Permutation
• Recurrence relation

Types of sequences

• Cauchy sequence
• Farey sequence
• Thue-Morse sequence
• Look-and-say sequence
• Fibonacci sequence
• Arithmetic progression
• Geometric progression

Related concepts

• List (computing)
• Recursion (computer science)
• Tuple
• Ordinal-indexed sequencehttp://en.wikipedia.org../../../../articles/o/r/d/Order_topology.html#Ordinal-indexed_sequences

Operations on sequences

• Limit of a sequence
• Cauchy product

External links

• The On-Line Encyclopedia of Integer Sequences
• Journal of Integer Sequences (free)
• Sequence on PlanetMath