|
|
| (699 intermediate revisions by more than 100 users not shown) |
| Line 1: |
Line 1: |
| {{Other uses}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
|
| |
|
| In [[mathematics]], a '''sequence''' is an ordered list of objects (or events). Like a [[Set (mathematics)|set]], it contains [[Element (mathematics)|members]] (also called ''elements'', or ''terms''). The number of ordered elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a [[Discrete mathematics|discrete]] [[function (mathematics)|function]].
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| | * Only registered users will be able to execute this rendering mode. |
| | * Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere. |
|
| |
|
| For example, (M, A, R, Y) is a sequence of letters that differs from (A, R, M, Y), as the ordering matters, and (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''[[finite set|finite]]'', as in this example, or ''[[Infinite set|infinite]]'', such as the sequence of all [[even and odd numbers|even]] [[positive and negative numbers|positive]] [[integer]]s (2, 4, 6,...). Finite sequences are sometimes known as ''strings'' or ''words'' and infinite sequences as ''streams''. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
| | Registered users will be able to choose between the following three rendering modes: |
| [[Image:Cauchy sequence illustration2.svg|right|thumb|350px|An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor [[Cauchy sequence|Cauchy]]. It is, however, bounded.]]
| |
|
| |
|
| == Examples and notation ==
| | '''MathML''' |
| 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.
| | :<math forcemathmode="mathml">E=mc^2</math> |
|
| |
|
| In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names for convenient referencing. For example a sequence might be written as (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, … ), or (''b''<sub>0</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>, … ), or (''c''<sub>0</sub>, ''c''<sub>2</sub>, ''c''<sub>4</sub>, … ), depending on what is useful in the application.
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
|
| |
|
| ===Finite and infinite===
| | '''source''' |
| A more formal definition of a '''finite sequence''' with terms in a set ''S'' is a [[function (mathematics)|function]] from {1, 2, ..., ''n''} to ''S'' for some ''n'' > 0. An '''infinite sequence''' in ''S'' is a function from {1, 2, ... } to ''S''. For example, the sequence of prime numbers (2,3,5,7,11, … ) is the function 1→'''2''', 2→'''3''', 3→'''5''', 4→'''7''', 5→'''11''', … .
| | :<math forcemathmode="source">E=mc^2</math> --> |
|
| |
|
| A sequence of a finite length ''n'' is also called an [[n-tuple|''n''-tuple]]. Finite sequences include the '''empty sequence''' ( ) that has no elements.
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
|
| |
|
| A function from ''all'' integers into a set is sometimes called a '''bi-infinite sequence''' or '''two-way infinite sequence'''. An example is the bi-infinite sequence of all even integers ( … , -4, -2, 0, 2, 4, 6, 8… ).
| | ==Demos== |
|
| |
|
| ===Multiplicative===
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| Let ''A'' = (a sequence defined by a function ''f'':{1, 2, 3, ...} → {1, 2, 3, ...}, such that ''a''<sub> ''i''</sub> = ''f''(''i'').
| |
|
| |
|
| The sequence is '''multiplicative''' if ''f''(''xy'') = ''f''(''x'')''f''(''y'') for all ''x'',''y'' such that ''x'' and ''y'' are [[coprime]].<ref>{{cite book|title=Lectures on generating functions|last=Lando|first=Sergei K.|publisher=AMS|ISBN=0-8218-3481-9|chapter=7.4 Multiplicative sequences}}</ref>
| |
|
| |
|
| ==Types and properties==
| | * accessibility: |
| 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.
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
|
| |
|
| If the terms of the sequence are a subset of an [[partially ordered set|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]]ly 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|monotonicity]] property is called monotonic or ''monotone''. This is a special case of the more general notion of [[monotonic function]].
| | ==Test pages == |
|
| |
|
| The terms ''nondecreasing'' and ''nonincreasing'' are used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively.
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
|
| |
|
| If the terms of a sequence are [[integer]]s, then the sequence is an [[integer sequence]]. If the terms of a sequence are [[polynomial]]s, then the sequence is a [[polynomial sequence]].
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| 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]].
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| If A is a set, the [[free monoid]] over A (denoted A<sup>*</sup>) is a [[monoid]] containing all the finite sequences (or strings) of zero or more elements drawn from A, with the binary operation of concatenation. The [[free semigroup]] A<sup>+</sup> is the subsemigroup of A<sup>*</sup> containing all elements except the empty sequence.
| |
| | |
| ==Analysis==
| |
| In [[mathematical analysis|analysis]], when talking about sequences, one will generally consider sequences of the form
| |
| :<math>(x_1, x_2, x_3, \dots)\text{ or }(x_0, x_1, x_2, \dots)\,</math>
| |
| which is to say, infinite sequences of elements indexed by [[natural number]]s.
| |
| | |
| It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by ''x<sub>n</sub>'' = 1/[[logarithm|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 number]]s.
| |
| This type can be generalized to sequences of elements of some [[vector space]]. In analysis, the vector spaces considered are often [[function space]]s. Even more generally, one can study sequences with elements in some [[topological space]].
| |
| | |
| ==Series== | |
| {{main|Series (mathematics)}}
| |
| The sum of terms of a sequence is a [[series (mathematics)|series]]. More precisely, if (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ...) is a sequence, one may consider the sequence of [[partial sum]]s (''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ...), with
| |
| | |
| :<math>S_n=x_1+x_2+\cdots + x_n=\sum\limits_{i=1}^{n}x_i.</math> | |
| | |
| Formally, this pair of sequences comprises the ''series'' with the terms ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ..., which is denoted as
| |
| | |
| :<math>\sum\limits_{i=1}^\infty x_i.</math>
| |
| | |
| If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see [[series (mathematics)|series]].
| |
| | |
| ==Infinite sequences in theoretical computer science== | |
| Infinite sequences of [[numerical digit|digits]] (or [[character (computing)|characters]]) drawn from a [[finite set|finite]] [[alphabet (computer science)|alphabet]] are of particular interest in [[theoretical computer science]]. They are often referred to simply as ''sequences'' or ''streams'', as opposed to finite ''[[String (computer science)#Formal theory|strings]]''. Infinite binary sequences, for instance, are infinite sequences of [[bit]]s (characters drawn from the alphabet {0,1}). The set ''C'' = {0, 1}<sup>∞</sup> 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 class]]es, 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.
| |
| | |
| == Vectors ==
| |
| Sequences over a field may also be viewed as [[Vector (geometric)|vectors]] in a [[vector space]]. Specifically, the set of ''F''-valued sequences (where ''F'' is a [[field (mathematics)|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 <math>\mathbb{C}</math>.
| |
| | |
| == {{anchor|Doubly infinite|Doubly infinite sequences|Doubly-infinite sequences}} 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 elements of the [[group ring|semigroup ring]] of the [[natural numbers]] <math>R[\N]</math>, and doubly infinite sequences as elements of the [[group ring]] of the [[integer]]s <math>R[\Z]</math>. This perspective is used in the [[Cauchy product]] of sequences.
| |
| | |
| ==Ordinal-indexed sequence==
| |
| An [[Order_topology#Ordinal-indexed_sequences|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.
| |
| | |
| == Sequences and automata ==
| |
| [[Automata theory|Automata]] or [[finite state machine]]s can typically be 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.
| |
| | |
| ===Types===
| |
| *[[±1-sequence]]
| |
| *[[Arithmetic progression]]
| |
| *[[Cauchy sequence]]
| |
| *[[Farey sequence]]
| |
| *[[Fibonacci number|Fibonacci sequence]]
| |
| *[[Geometric progression]]
| |
| *[[Look-and-say sequence]]
| |
| *[[Thue–Morse sequence]]
| |
| | |
| ===Related concepts===
| |
| *[[List (computing)]]
| |
| *[[Order_topology#Ordinal-indexed_sequences|Ordinal-indexed sequence]]
| |
| *[[Recursion (computer science)]]
| |
| *[[Tuple]]
| |
| *[[Set theory]]
| |
| | |
| ===Operations===
| |
| *[[Cauchy product]]
| |
| *[[Limit of a sequence]]
| |
| | |
| ==See also==
| |
| *[[Set (mathematics)]]
| |
| *[[Net (topology)]] (a generalization of sequences)
| |
| *[[On-Line Encyclopedia of Integer Sequences]]
| |
| *[[Permutation]]
| |
| *[[Recurrence relation]]
| |
| *[[Sequence space]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==External links==
| |
| *[http://oeis.org/ The On-Line Encyclopedia of Integer Sequences]
| |
| *[http://www.cs.uwaterloo.ca/journals/JIS/index.html Journal of Integer Sequences] (free)
| |
| *{{planetmath reference|id=397|title=Sequence}}
| |
| | |
| [[Category:Elementary mathematics]]
| |
| [[Category:Sequences and series|*]]
| |
| | |
| [[ar:متتالية]]
| |
| [[bg:Редица]]
| |
| [[bs:Niz]]
| |
| [[ca:Successió (matemàtiques)]]
| |
| [[cs:Posloupnost]]
| |
| [[da:Talfølge]]
| |
| [[de:Folge (Mathematik)]]
| |
| [[et:Jada]]
| |
| [[el:Ακολουθία]]
| |
| [[es:Sucesión matemática]]
| |
| [[eo:Vico]]
| |
| [[eu:Segida (matematika)]]
| |
| [[fa:دنباله]]
| |
| [[fr:Suite (mathématiques)]]
| |
| [[gl:Sucesión (matemáticas)]]
| |
| [[xal:Даралт]]
| |
| [[ko:수열]]
| |
| [[hy:Հաջորդականություն (մաթեմատիկական)]]
| |
| [[hr:Niz]]
| |
| [[io:Sequo]]
| |
| [[is:Runa]]
| |
| [[it:Successione (matematica)]]
| |
| [[he:סדרה]]
| |
| [[ka:მიმდევრობა]]
| |
| [[kk:Іштізбек]]
| |
| [[la:Sequentia (mathematica)]]
| |
| [[hu:Sorozat (matematika)]]
| |
| [[mk:Низа (математика)]]
| |
| [[ml:അനുക്രമം]]
| |
| [[ms:Jujukan]]
| |
| [[nl:Rij (wiskunde)]]
| |
| [[ja:列 (数学)]]
| |
| [[no:Følge (matematikk)]]
| |
| [[nn:Følgje]]
| |
| [[pms:Sequensa]]
| |
| [[pl:Ciąg (matematyka)]]
| |
| [[pt:Sequência (matemática)]]
| |
| [[ro:Șir (matematică)]]
| |
| [[ru:Последовательность]]
| |
| [[scn:Succissioni (matimatica)]]
| |
| [[simple:Sequence]]
| |
| [[sk:Postupnosť (matematika)]]
| |
| [[sl:Zaporedje]]
| |
| [[sr:Низ]]
| |
| [[fi:Lukujono]]
| |
| [[sv:Följd]]
| |
| [[ta:தொடர்வரிசை]]
| |
| [[th:ลำดับ]]
| |
| [[tr:Dizi (terim)]]
| |
| [[uk:Послідовність (математика)]]
| |
| [[ur:متوالیہ (ریاضی)]]
| |
| [[vi:Dãy (toán học)]]
| |
| [[zh:序列]]
| |
