SequenceFrom Wikipedia, the free encyclopediaSequential redirects here. For the manual transmission, see Sequential manual transmission. For other uses, seeSequence (disambiguation).In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it containsmembers (also called elements, or terms). The number of elements (possibly innite) is called the length of thesequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at dierent positionsin the sequence. Formally, a sequence can be dened as a function whose domain is a countable totally ordered set,such as the natural numbers.For example, (M, A, R, Y) is a sequence of letters with the letter 'M' rst and 'Y' last.This sequence diers from(A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two dierent positions, is avalid sequence. Sequences can be nite, as in these examples, or innite, such as the sequence of all even positiveintegers (2, 4, 6,...). In computing and computer science, nite sequences are sometimes called strings, words or lists,the dierent names commonly corresponding to dierent ways to represent them into computer memory; innitesequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may beexcluded depending on the context.An innite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. Itis, however, bounded.1 Examples and notationA sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number ofmathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence12 1 EXAMPLES AND NOTATIONproperties of sequences. In particular, sequences are the basis for series, which are important in dierential equationsand analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in thestudy of prime numbers.There are a number of ways to denote a sequence, some of which are more useful for specic types of sequences.One way to specify a sequence is to list the elements. For example, the rst four odd numbers form the sequence(1,3,5,7). This notation can be used for innite sequences as well. For instance, the innite sequence of positiveodd integers can be written (1,3,5,7,...). Listing is most useful for innite sequences with a pattern that can be easilydiscerned from the rst few elements. Other ways to denote a sequence are discussed after the examples.1.1 Important examples321 158A tiling with squares whose sides are successive Fibonacci numbers in length.There are many important integer sequences. The prime numbers are the natural numbers bigger than 1, that haveno divisors but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). Thestudy of prime numbers has important applications for mathematics and specically number theory.The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The rsttwo elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...).Other interesting sequences include the ban numbers, whose spellings do not contain a certain letter of the alphabet.For instance, the eban numbers (do not contain 'e') formthe sequence (2,4,6,30,32,34,36,40,42,...). Another sequencebased on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...).For a list of important examples of integers sequences see On-line Encyclopedia of Integer Sequences.Other important examples of sequences include ones made up of rational numbers, real numbers, and complex num-bers. The sequence (.9,.99,.999,.9999,...) approaches the number 1. In fact, every real number can be written asthe limit of a sequence of rational numbers. It is this fact that allows us to write any real number as the limit of asequence of decimals. For instance, is the limit of the sequence (3,3.1,3.14,3.141,3.1415,...). The sequence for ,however, does not have any pattern that is easily discernible by eye, unlike the sequence (0.9,0.99,...).1.2 Indexing 31.2 IndexingOther notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not havea pattern such as the digits of . This section focuses on the notations used for sequences that are a map from a subsetof the natural numbers. For generalizations to other countable index sets see the following section and below.The terms of a sequence are commonly denoted by a single variable, say an, where the index n indicates the nthelement of the sequence.a1 element 1sta2 element 2nda3 element 3rd......an1 element (n-1)than element nthan+1 element (n+1)th......Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whoseelements are related to the index n (the elements position) in a simple way. For instance, the sequence of the rst 10square numbers could be written as(a1, a2, ..., a10), ak= k2.This represents the sequence (1,4,9,...100). This notation is often simplied further as(ak)10k=1, ak= k2.Here the subscript {k=1} and superscript 10 together tell us that the elements of this sequence are the ak such that k= 1, 2, ..., 10.Sequences can be indexed beginning and ending from any integer. The innity symbol is often used as the super-script to indicate the sequence including all integer k-values starting with a certain one. The sequence of all positivesquares is then denoted(ak)k=1, ak= k2.In cases where the set of indexing numbers is understood, such as in analysis, the subscripts and superscripts are oftenleft o. That is, one simply writes ak for an arbitrary sequence. In analysis, k would be understood to run from 1 to. However, sequences are often indexed starting from zero, as in(ak)k=0= (a0, a1, a2, ...).In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easilyinferred. In these cases the index set may be implied by a listing of the rst few abstract elements. For instance, thesequence of squares of odd numbers could be denoted in any of the following ways. (1, 9, 25, ...) (a1, a3, a5, ...), ak= k2 (a2k1)k=1, ak= k24 2 FORMAL DEFINITION AND BASIC PROPERTIES (ak)k=1, ak= (2k 1)2 ((2k 1)2)k=1Moreover, the subscripts and superscripts could have been left o in the third, fourth, and fth notations if theindexing set was understood to be the natural numbers.Finally, sequences can more generally be denoted by writing a set inclusion in the subscript, such as in(ak)kNThe set of values that the index can take on is called the index set. In general, the ordering of the elements ak isspecied by the order of the elements in the indexing set. When N is the index set, the element ak+1 comes after theelement ak since in N, the element (k+1) comes directly after the element k.1.3 Specifying a sequence by recursionSequences whose elements are related to the previous elements in a straightforward way are often specied usingrecursion. This is in contrast to the specication of sequence elements in terms of their position.To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones beforeit. In addition, enough initial elements must be specied so that new elements of the sequence can be specied bythe rule. The principle of mathematical induction can be used to prove that a sequence is well-dened, which is tosay that that every element of the sequence is specied at least once and has a single, unambiguous value. Inductioncan also be used to prove properties about a sequence, especially for sequences whose most natural specication isby recursion.The Fibonacci sequence can be dened using a recursive rule along with two initial elements. The rule is that eachelement is the sum of the previous two elements, and the rst two elements are 0 and 1.an= an1 + an2 , with a0= 0 and a1= 1 .The rst ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence thatis dened recursively is Recamans sequence, considered at the beginning of this section. We can dene Recamanssequence bya0= 0 and an= an1n if the result is positive and not already in the list. Otherwise, an= an1+n.Not all sequences can be specied by a rule in the form of an equation, recursive or not, and some can be quitecomplicated. For example, the sequence of prime numbers is the set of prime numbers in their natural order. Thisgives the sequence (2,3,5,7,11,13,17,...).One can also notice that the next element of a sequence is a function of the element before, and so we can write thenext element as :an+1= f(an)This functional notation can prove useful when one wants to prove the global monotony of the sequence.2 Formal denition and basic propertiesThere are many dierent notions of sequences in mathematics, some of which (e.g., exact sequence) are not coveredby the denitions and notations introduced below.2.1 Formal denitionA sequence is usually dened as a function whose domain is a countable totally ordered set, although in many disci-plines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset2.2 Finite and innite 5of the natural numbers to the real numbers.[1] In other words, a sequence is a map f(n) : N R. To recover ourearlier notation we might identify an = f(n) for all n or just write an : N R.In complex analysis, sequences are dened as maps from the natural numbers to the complex numbers (C).[2] Intopology, sequences are often dened as functions from a subset of the natural numbers to a topological space.[3]Sequences are an important concept for studying functions and, in topology, topological spaces. An important gener-alization of sequences, called a net, is to functions from a (possibly uncountable) directed set to a topological space.2.2 Finite and inniteThe length of a sequence is dened as the number of terms in the sequence.A sequence of a nite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has noelements.Normally, the term innite sequence refers to a sequence which is innite in one direction, and nite in the otherthesequence has a rst element, but no nal element, it is called a singly innite, or one-sided (innite) sequence,when disambiguation is necessary. In contrast, a sequence that is innite in both directionsi.e. that has neither arst nor a nal elementis called a bi-innite sequence, two-way innite sequence, or doubly innite sequence.A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( , 4, 2,0, 2, 4, 6, 8 ), is bi-innite. This sequence could be denoted (2n)n= .One can interpret singly innite sequences as elements of the semigroup ring of the natural numbers R[N], and doublyinnite sequences as elements of the group ring of the integers R[Z]. This perspective is used in the Cauchy productof sequences.2.3 Increasing and decreasingA sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For asequence(an)n=1 this can be written as an an for all n N. If each consecutive term is strictly greater than(>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonicallydecreasing if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasingif each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotonesequence. This is a special case of the more general notion of a monotonic function.The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoidany possible confusion with strictly increasing and strictly decreasing, respectively.2.4 BoundedIf the sequence of real numbers (an) is such that all the terms, after a certain one, are less than some real number M,then the sequence is said to be bounded from above. In less words, this means an M for all n greater than N forsome pair M and N. Any such M is called an upper bound.Likewise, if, for some real m, an m for all n greaterthan some N, then the sequence is bounded from below and any such m is called a lower bound.If a sequence isboth bounded from above and bounded from below then the sequence is said to be bounded.2.5 Other types of sequencesA subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elementswithout disturbing the relative positions of the remaining elements. For instance, the sequence of positive evenintegers (2,4,6,...) is a subsequence of the positive integers (1,2,3,...). The positions of some elements change whenother elements are deleted. However, the relative positions are preserved.Some other types of sequences that are easy to dene include:An integer sequence is a sequence whose terms are integers.A polynomial sequence is a sequence whose terms are polynomials.6 3 LIMITS AND CONVERGENCEA positive integer sequence is sometimes called multiplicative if anm = an am for all pairs n,m such that n andm are coprime.[4] In other instances, sequences are often called multiplicative if an = na1 for all n. Moreover,the multiplicative Fibonacci sequence satises the recursion relation an = an an.3 Limits and convergenceMain article: Limit of a sequenceOne of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit.5 10 15 20 250.00.20.40.60.81.0n + 12n2The plot of a convergent sequence (a) is shown in blue. Visually we can see that the sequence is converging to the limit zero as nincreases.Continuing informally, a (singly innite) sequence has a limit if it approaches some value L, called the limit, as nbecomes very large. That is, for an abstract sequence (an) (with n running from 1 to innity understood) the valueof an approaches L as n , denotedlimnan= L.More precisely, the sequence converges if there exists a limit L such that the remaining a's are arbitrarily close to Lfor some n large enough.If a sequence converges to some limit, then it is convergent; otherwise it is divergent.If an gets arbitrarily large as n we writelimnan= .3.1 Denition of convergence 7In this case we say that the sequence (an) diverges, or that it converges to innity.If an becomes arbitrarily small negative numbers (large in magnitude) as n we writelimnan= and say that the sequence diverges or converges to minus innity.3.1 Denition of convergenceFor sequences that can be written as (an)n=1 with an R we can write (an) with the indexing set understood as N.These sequences are most common in real analysis. The generalizations to other types of sequences are consideredin the following section and the main page Limit of a sequence.Let (an) be a sequence. In words, the sequence (an) is said to converge if there exists a number L such that no matterhow close we want the an to be to L (say -close where > 0), we can nd a natural number N such that all terms(aN+1, aN+2, ...) are further closer to L (within of L).[1] This is often written more compactly using symbols. Forinstance,for all > 0, there exists a natural number N such that L < an < L+ for all n N.In even more compact notation > 0, N N s.t. n N, |an L| < .The dierence in the denitions of convergence for (one-sided) sequences in complex analysis and metric spaces isthat the absolute value |an L| is interpreted as the distance in the complex plane ( zz ), and the distance underthe appropriate metric, respectively.3.2 Applications and important resultsImportant results for convergence and limits of (one-sided) sequences of real numbers include the following. Theseequalities are all true at least when both sides exist. For a discussion of when the existence of the limit on one sideimplies the existence of the other see a real analysis text such as can be found in the references.[1][5]The limit of a sequence is unique.limn(an bn) = limnan limnbnlimncan= c limnanlimn(anbn) = (limnan)(limnbn)limnanbn=limnanlimnbnprovided limnbn = 0limnapn= [limnan]pIf an bn for all n greater than some N, then limnan limnbn .(Squeeze Theorem) If an cn bn for all n > N, and limnan= limnbn= L, then limncn= L.If a sequence is bounded and monotonic then it is convergent.A sequence is convergent if and only if every subsequence is convergent.8 4 SERIESThe plot of a Cauchy sequence (X), shown in blue, as X versus n. Visually, we see that the sequence appears to be converging to thelimit zero as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence convergesto some limit.3.3 Cauchy sequencesMain article: Cauchy sequenceA Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion ofa Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. Oneparticularly important result in real analysis is Cauchy characterization of convergence for sequences:In the real numbers, a sequence is convergent if and only if it is Cauchy.In contrast, in the rational numbers, e.g. the sequence dened by x1 = 1 and xn = xn + 2/xn/2 is Cauchy, but has norational limit, cf. here.4 SeriesMain article: Series (mathematics)A series is, informally speaking, the sum of the terms of a sequence. That is, adding the rst N terms of a (one-sided)sequence forms the Nth term of another sequence, called a series. Thus the N series of the sequence (a) results inanother sequence (SN) given by:S1= a1S2= a1 + a2S3= a1 + a2+ a3......SN= a1 + a2+ a3 + ......We can also write the nth term of the series as9SN=Nn=1an.Then the concepts used to talk about sequences, such as convergence, carry over to series (the sequence of partialsums) and the properties can be characterized as properties of the underlying sequences (such as (an) in the lastexample). The limit, if it exists, of an innite series (the series created from an innite sequence) is written aslimNSN=n=1an.5 Use in other elds of mathematics5.1 TopologySequence play an important role in topology, especially in the study of metric spaces. For instance:A metric space is compact exactly when it is sequentially compact.Afunction froma metric space to another metric space is continuous exactly when it takes convergent sequencesto convergent sequences.A metric space is a connected space if, whenever the space is partitioned into two sets, one of the two setscontains a sequence converging to a point in the other set.A topological space is separable exactly when there is a dense sequence of points.Sequences can be generalized to nets or lters. These generalizations allow one to extend some of the above theoremsto spaces without metrics.5.1.1 Product topologyA product space of a sequence of topological spaces is the cartesian product of the spaces equipped with a naturaltopology called the product topology.More formally, given a sequence of spaces {Xi} , dene X such thatX:=iIXi,is the set of sequences {xi} where each xi is an element of Xi . Let the canonical projections be written as pi :X Xi. Then the product topology on X is dened to be the coarsest topology (i.e. the topology with the fewestopen sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonotopology.5.2 AnalysisIn analysis, when talking about sequences, one will generally consider sequences of the form(x1, x2, x3, . . . ) or (x0, x1, x2, . . . )which is to say, innite sequences of elements indexed by natural numbers.10 5 USE IN OTHER FIELDS OF MATHEMATICSIt may be convenient to have the sequence start with an index dierent from1 or 0. For example, the sequence denedby xn = 1/log(n) would be dened only for n 2. When talking about such innite sequences, it is usually sucient(and does not change much for most considerations) to assume that the members of the sequence are dened at leastfor 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 typecan be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are oftenfunction spaces. Even more generally, one can study sequences with elements in some topological space.5.2.1 Sequence spacesMain article: Sequence spaceAsequence space is a vector space whose elements are innite sequences of real or complex numbers. Equivalently, itis a function space whose elements are functions from the natural numbers to the eld K of real or complex numbers.The set of all such functions is naturally identied with the set of all possible innite sequences with elements in K,and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalarmultiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped witha norm, or at least the structure of a topological vector space.The most important sequences spaces in analysis are the pspaces, consisting of the p-power summable sequences,with the p-norm. These are special cases of Lpspaces for the counting measure on the set of natural numbers.Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectivelydenoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwiseconvergence, under which it becomes a special kind of Frchet space called FK-space.5.3 Linear algebraSequences over a eld may also be viewed as vectors in a vector space. Specically, the set of F-valued sequences(where F is a eld) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.5.4 Abstract algebraAbstract algebra employs several types of sequences, including sequences of mathematical objects such as groups orrings.5.4.1 Free monoidMain article: Free monoidIf A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the nitesequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroupA+ is the subsemigroup of A* containing all elements except the empty sequence.5.4.2 Exact sequencesMain article: Exact sequenceIn the context of group theory, a sequenceG0f1 G1f2 G2f3 fn Gnof groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to thekernel of the next:5.5 Set theory 11im(fk) = ker(fk+1)Note that the sequence of groups and homomorphisms may be either nite or innite.Asimilar denition can be made for certain other algebraic structures. For example, one could have an exact sequenceof vector spaces and linear maps, or of modules and module homomorphisms.5.4.3 Spectral sequencesMain article: Spectral sequenceIn homological algebra and algebraic topology, aspectralsequence is a means of computing homology groupsby taking successive approximations. Spectral sequences are a generalization of exact sequences, and since theirintroduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.5.5 Set theoryAn ordinal-indexed sequence is a generalization of a sequence. If is a limit ordinal and X is a set, an -indexedsequence of elements of X is a function from to X. In this terminology an -indexed sequence is an ordinarysequence.5.6 ComputingAutomata or nite state machines can typically be thought of as directed graphs, with edges labeled using some specicalphabet, . Most familiar types of automata transition from state to state by reading input letters from , followingedges 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 nondeterministicautomaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some inputletter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence ofsingle states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally usedto mean the latter.5.7 StreamsInnite sequences of digits (or characters) drawn from a nite alphabet are of particular interest in theoretical com-puter science. They are often referred to simply as sequences or streams, as opposed to nite strings. Innite binarysequences, for instance, are innite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0,1} of all innite, binary sequences is sometimes called the Cantor space.An innite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalizationmethod for proofs.[6]6 Types1-sequenceArithmetic progressionCauchy sequenceFarey sequenceFibonacci sequence12 10 REFERENCESGeometric progressionLook-and-say sequenceThueMorse sequence7 Related conceptsList (computing)Ordinal-indexed sequenceRecursion (computer science)TupleSet theory8 OperationsCauchy productLimit of a sequence9 See alsoEnumerationNet (topology) (a generalization of sequences)On-Line Encyclopedia of Integer SequencesPermutationRecurrence relationSequence spaceSet (mathematics)10 References[1] Gaughan, Edward. 1.1 Sequences and Convergence. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.[2] Edward B. Sa & Arthur David Snider (2003). Chapter 2.1. Fundamentals of Complex Analysis. ISBN 01-390-7874-6.[3] James R. Munkres. Chapters 1&2. Topology. ISBN 01-318-1629-2.[4] Lando, Sergei K. 7.4 Multiplicative sequences. Lectures on generating functions. AMS. ISBN 0-8218-3481-9.[5] Dawikins, Paul. Series and Sequences. Pauls Online Math Notes/Calc II (notes). Retrieved 18 December 2012.[6] Oazer, Kemal. FORMAL LANGUAGES, AUTOMATAANDCOMPUTATION: DECIDABILITY (PDF). cmu.edu.Carnegie-Mellon University. Retrieved 24 April 2015.1311 External linksThe dictionary denition of sequence at WiktionaryHazewinkel, Michiel, ed. (2001), Sequence, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4The On-Line Encyclopedia of Integer SequencesJournal of Integer Sequences (free)Sequence at PlanetMath.org.14 12 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES12 Text and image sources, contributors, and licenses12.1 Text Sequence Source:https://en.wikipedia.org/wiki/Sequence?oldid=675741230 Contributors:AxelBoldt, Mav, Zundark, Tarquin, XJaM,Toby Bartels, Imran, Camembert, Youandme, Lir, Patrick, Michael Hardy, Ihcoyc, Poor Yorick, Nikai, EdH, Charles Matthews, Dys-prosia, Greenrd, Hyacinth, Zero0000, Sabbut, Garo, Robbot, Lowellian, MathMartin, Stewartadcock, Henrygb, Bkell, Tosha, Centrx,Giftlite, BenFrantzDale, Lupin, Herbee, Horatio, Edcolins, Vadmium, Leonard Vertighel, Manuel Anastcio, Alexf, Fudo, Melikamp,Sam Hocevar, Tsemii, Ross bencina, Jiy, TedPavlic, Paul August, JoeSmack, Elwikipedista~enwiki, Syp, Pjrich, Shanes, Jonathan Drain,Nk, Obradovic Goran, Haham hanuka, Zaraki~enwiki, Merope, Jumbuck, Reubot, Jet57, Olegalexandrov, Ringbang, Djsasso, Total-cynic, Oleg Alexandrov, Hoziron, Linas, Madmardigan53, MFH, Isnow, Graham87, Dpv, Mendaliv, Salix alba, Figs, VKokielov, Loggie,Rsenington, RexNL, Pexatus, Fresheneesz, Kri, Ryvr, Chobot, Lightsup55, Krishnavedala, Wavelength, Michael Slone, Grubber, ArthurRubin, JahJah, Pred, Finell, KHenriksson, Gelingvistoj, Chris the speller, Bluebot, Nbarth, Mcaruso, Suicidalhamster, SundarBot, Dread-star, Fagstein, Just plain Bill, Xionbox, Dreftymac, Gco, CRGreathouse, CBM, Gregbard, Cydebot, Xantharius, Epbr123, KClier, SaberCherry, Rlupsa, Marek69, Urdutext, Icep, Ste4k, Mutt Lunker, JAnDbot, Asnac, Coolhandscot, Martinkunev, Magioladitis, VoABot II,Avjoska, JamesBWatson, Brusegadi, Minimiscience, Stdazi, DerHexer, J.delanoy, Trusilver, Suenm~enwiki, Ncmvocalist, Belovedfreak,Policron, JingaJenga, VolkovBot, ABF, AlnoktaBOT, Philip Trueman, Digby Tantrum, JhsBot, Isis4563, Wolfrock, Xiong Yingfei, New-byguesses, SieBot, Scarian, Yintan, Xelgen, Outs, Paolo.dL, OKBot, Pagen HD, Wahrmund, Classicalecon, Atif.t2, Crambo0349, Clue-Bot, Justin W Smith, Fyyer, SuperHamster, Excirial, Estirabot, Jotterbot, Thingg, Downgrader, Aj00200, XLinkBot, Stickee, Rror, Wik-Head, Brentsmith101, Addbot, Non-dropframe, Kongr43gpen, Matj Grabovsk, Legobot, Luckas-bot, Yobot, Eric-Wester, 4th-otaku,AnomieBOT, Jim1138, Law, Materialscientist, E2eamon, ArthurBot, Ayda D, Xqbot, Omnipaedista, RibotBOT, Charvest, Shadow-jams, Thehelpfulbot, Dan6hell66, Constructive editor, Mark Renier, Tal physdancer, SixPurpleFish, Pinethicket, BRUTE, SkyMachine,PiRSquared17, Roy McCoy, RjwilmsiBot, Tzfyr, EmausBot, John of Reading, GoingBatty, Wikipelli, K6ka, Brent Perreault, Nelland-mice, Bethnim, Ida Shaw, Alpha Quadrant, KuduIO, D.Lazard, SporkBot, Wayne Slam, Donner60, Chewings72, ClueBot NG, Satel-lizer, Widr, MerlIwBot, Helpful Pixie Bot, HMSSolent, Curb Chain, Calabe1992, Brad7777, Minsbot, Praxiphenes, EuroCarGT, VenSeyranyan., Jegyao, DavyRalph, Graphium, Jochen Burghardt, Brirush, Mark viking, LoMaPh, Immonster, EricsonWillians, Emlynlee,Buscus 3, JackHoang, BemusedObserver, Some1Redirects4You and Anonymous: 21012.2 Images File:Cauchy_sequence_illustration.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/62/Cauchy_sequence_illustration.svg License: CC0 Contributors: Own work Original artist: Krishnavedala File:Cauchy_sequence_illustration2.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7a/Cauchy_sequence_illustration2.svg License: Public domain Contributors: Based on File:Cauchy_sequence_illustration2.png by Oleg Alexandrov Original artist: Own work File:Converging_Sequence_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e4/Converging_Sequence_example.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Melikamp File:Fibonacci_blocks.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/bc/Fibonacci_blocks.svg License: Public do-main Contributors: Own work Original artist: ElectroKid ( ) File:Fibonacci_spiral_34.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/93/Fibonacci_spiral_34.svg License: Publicdomain Contributors: self-drawn in Inkscape Original artist: User:Dicklyon12.3 Content license Creative Commons Attribution-Share Alike 3.0