# Dictionary Definition

sequential adj : in regular succession without
gaps; "serial concerts" [syn: consecutive, sequent, serial, successive]

# User Contributed Dictionary

## English

### Adjective

- Succeeding or following in order.

#### Derived terms

#### Translations

- Finnish: peräkkäinen, peräkkäis-

# Extensive Definition

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, as in
this example, or infinite,
such as the sequence of all even
positive 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
from to S for some n ≥ 0. An infinite sequence in S is a function
from (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 sequence of a fixed-length n 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
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.

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.

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 analysis, when talking about sequences, one will generally consider sequences of the form- (x_1, x_2, x_3, \dots)\, or (x_0, x_1, x_2, \dots)\,

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/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.

## Series

The sum of terms of a sequence is a series. 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_^x_i.

- \sum\limits_^x_i.

## 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 ). The set C = ∞ 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 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 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.In particular, the term sequence
space usually refers to a linear
subspace of the set of all possible infinite sequences with
elements in \mathbb.

## 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 R[\N], and doubly infinite sequences as elements of the
group
ring of the integers
R[\Z]. 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 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

### Related concepts

### Operations on sequences

## External links

sequential in Arabic: متتالية

sequential in Bulgarian: Редица

sequential in Catalan: Successió
matemàtica

sequential in Danish: Talfølge

sequential in German: Folge (Mathematik)

sequential in Modern Greek (1453-):
Ακολουθία

sequential in Spanish: Sucesión matemática

sequential in Esperanto: Vico

sequential in Persian: توالی

sequential in French: Suite
(mathématiques)

sequential in Galician: Sucesión
(matemáticas)

sequential in Korean: 수열

sequential in Croatian: Niz

sequential in Ido: Sequo

sequential in Indonesian: Barisan

sequential in Icelandic: Runa

sequential in Italian: Successione
(matematica)

sequential in Hebrew: סדרה

sequential in Hungarian: Sorozat
(matematika)

sequential in Dutch: Rij (wiskunde)

sequential in Japanese: 列 (数学)

sequential in Norwegian: Følge

sequential in Polish: Ciąg (matematyka)

sequential in Portuguese: Seqüência
matemática

sequential in Romanian: Şir (matematică)

sequential in Russian: Последовательность

sequential in Sicilian: Succissioni
(matimatica)

sequential in Simple English: Sequence

sequential in Slovenian: Zaporedje

sequential in Finnish: Lukujono

sequential in Swedish: Följd

sequential in Tamil: தொடர்வரிசை

sequential in Thai: ลำดับ

sequential in Vietnamese: Dãy (toán học)

sequential in Ukrainian: Послідовність
(математика)

sequential in Urdu: متوالیہ (ریاضی)

sequential in Chinese: 序列

# Synonyms, Antonyms and Related Words

appendant, catenary, connected, consecutive, consequent, consequential, consistent, continuous, cyclic, deducible, derivable, derivational, derivative, ensuing, final, following, infenible, joined, lineal, linear, next, ordered, orderly, ordinal, organized, posterior, postpositional, postpositive, progressive, proximate, resultant, resulting, sequacious, sequent, serial, seriate, subsequent, succeeding, successional, successive, suffixed, systematic, tight, uninterrupted