Sequence

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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.

For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite set, as in this example, or infinity, such as the sequence of all even and odd numbers positive and negative numbers integers (2, 4, 6,...).

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.

A sequence may be denoted (a1, a2, ...). For shortness, the notation (an) is also used.

A more formal definition of a finite sequence with terms in a set S is a function (mathematics) 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.

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

A finite sequence is also called an n-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.

If the terms of the sequence are a subset of an partially 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 strict greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonic function 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 aninteger sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence.

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.

Sequences in analysis In mathematical analysis, when talking about sequences, one will generally consider sequences of the form (x_1, x_2, x_3, ...)\, or (x_0, x_1, x_2, ...)\, 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 xn = 1/logarithm(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.

Series The sum of terms of a sequence is a series (mathematics). More precisely, if (x1, x2, x3, ...) is a sequence, one may consider the sequence of partial sums (S1, S2, S3, ...), with S_n=x_1+x_2+\dots + x_n=\sum\limits_{i=1}^{n}x_i. Formally, this pair of sequences comprises the series with the terms x1, x2, x3, ..., which is denoted as \sum\limits_{i=1}^{\infty}x_i. If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see series (mathematics).

Infinite sequences in theoretical computer science Infinite sequences of numerical digit (or character (computing)) drawn from a finite set alphabet (computer science) are of particular interest in theoretical computer science. They are often referred to simply as sequences (as opposed to finite String (computer science)#Formal theory). 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.

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.

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.

Sequences as vectors Sequences over a field may also be viewed as vector (spatial) in a vector space. Specifically, the set of F-valued sequences (where F is a field (mathematics)) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

In particular, the term sequence space usually refers to a linear subspace of the set of all possible infinite sequences with elements in \mathbb{C}.

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 group ring of the natural numbers R, and doubly infinite sequences as elements of the group ring of the integers R. This perspective is used in the Cauchy product of sequences.

Ordinal-indexed sequence An 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.

Sequences and automata Automata_theory 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 number
Arithmetic progression
Geometric progression

Related concepts

List (computing)
Tuple

Operations on sequences

Limit of a sequence
Cauchy product

External links

The On-Line Encyclopedia of Integer Sequences
Journal of Integer Sequences. (free)

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