Talk:Sequence/Archive 1

The biochem definition of a sequence is really a special case of the mathematical one, isn't it? Maybe the info there would be better given on polymer or biopolymer?

This article needs a complete rewrite. Some is ambiguous, or wrong, or tries to be finite and infinite at the same time, etc.. --Zero 22:48, 31 Mar 2004 (UTC)

Proposed major revision

This article is a mess. I propose to replace the first 3/4 of it with the following text. The main difference is that the formal definition is given more carefully and finite sequences are not treated as an afterthought. Any objections?

Start:

In mathematics, a sequence is a list of items that are arranged in a linear fashion like the natural numbers. For example, (C,Y,R) is a sequence of letters; the ordering is that C is first, Y is second, and R is third. Sequences can be finite, as in the example just given, or infinite, such as the sequence of all even positive integers (2,4,6,...). Finite sequences include the null sequence ( ) that has no elements. The elements in a sequence are also called terms, and the number of terms (possibly infinite) is called the length of the sequence.

A more formal definition is that a finite sequence over a set S is an function from {1,2,...,n} to S for some n≥0. An infinite sequence over S is an function from {1,2,...} (the set of natural numbers) to S. If f is such a function, then in the previous notation the sequence is (f(1), f(2), ... ).

In modern mathematical terminology, the position (index) of a term is often indicated with a subscript, and (a_i) is a shorthand for the sequence (a₁,a₂, ...). If it is desired to indicate the range of values i takes, notations such as (a_i)⁵_i=1 can be used.

If S is the set of integers, then the sequence is an integer sequence. If S is a set of polynomials, the sequence is a polynomial sequence.

If S is endowed with a topology then it is possible to talk about convergence of an infinite sequence over S. This is discussed in detail in the article about limits.

A subsequence of a sequence S is a sequence formed from S by deleting some of the elements without disturbing the relative positions of the remaining elements.

:End (unsigned comment by Zero)

I can tell you why a finite sequence is an afterthought. If you noticed, this article is in the category "Calculus", and in there, people use exclusively infinite sequences. In most areas of mathematics a sequence is also infinite by default. A finite sequence is rather thought of an element of some Rⁿ, that is, as a vector.

Putting too much emphasis on finite sequences would be confusing. Again, we are talking calculus here.

I have a suggestion. Maybe you could write an article exclusively on finite sequences, with properties especially for them, etc. Then we can link there from the main article. What do you think?

PS Any calculus person would die, when hearing that, as you wrote, that there exist empty sequences.Oleg Alexandrov 03:22, 1 Jan 2005 (UTC)

There even exists an article for finite sequences, it is called N-tuple.

Thank you for not modifying the Sequence article right away, and for asking my feedback. I am now looking forward from feedback from you about how to make anybody happy as far as sequences are concerened. Oleg Alexandrov 04:04, 1 Jan 2005 (UTC)

The category is chosen to match the article. It is not a definition of what should be in the article. This article is about the mathematical concept of "sequence", which can be finite or infinite. Finite sequences are used everywhere in mathematics as a basic concept, including in calculus. Look at a book on multivariate calculus and you will see finite sequences on every page, so I don't think your comment about calculus is correct. --Zero 04:08, 1 Jan 2005 (UTC)

Ok, let me explain in different language. Finite sequences are used in math. But, the name "sequence" is used for infinite sequences only. What you see in a calculus book is called a vector. In combinatorics it is called a n-tuple.

I am just curious, what is your background? Oleg Alexandrov 04:15, 1 Jan 2005 (UTC)

Check your email ;-). --Zero 06:03, 1 Jan 2005 (UTC)

What you write is partly correct, in the sense that there are many contexts in which "sequence" is used to mean "infinite sequence" without ambiguity. For example, any context which involves convergence. However, it is also easy to find examples of the opposite. You can open a graph theory book and read "a path is a sequence of vertices such that..", or a group theory book and read "a composition series is a sequence of subgroups such that..." (both examples are finite in the case that the graph/group is finite but can be finite even if the graph/group is infinite). Wikipedia articles using "sequence" for finite mathematical objects include Goodstein's theorem, Farey sequence, Gray code, Viterbi algorithm, Merkle-Hellman, and probably others. Few people would use "n-tuple" for any of these except maybe the last two. --Zero 06:03, 1 Jan 2005 (UTC)

OK. This is a cultural clash. I am an analyst (well, applied mathematician). You see, the sequence article is heavily biased towards analysis. I mean, all that stuff discussed there, like convergence, monotonicity, series, polynomial sequences, are all about Calculus. I don't agree the article is badly written, I wrote it with a calculus perspective in mind.

Now, I see your point. A sequence can mean many things. All right, go ahead, try to modify the article, and please put lots of thoughts to make things clear. I don't want math rigor in an encycopidia, and I don't want the most general comprehensive definition either, if that in any way obscures the point or confuses things. This article was a mess before I got to it (just check older versions). So, I am looking forward to seeing what you've got to write. You need to make both the calculus people happy (like me), and the discrete people happy, and most importantly, the general public happy. And you do have to say something about n-tuples, because ultimately that's a finite sequence with a different name (see n-tuples article). And think about what's a good category to put the article too (besides calculus). Happy New Year and talk to you tomorrow. Oleg Alexandrov 06:19, 1 Jan 2005 (UTC)

proposed modifications

1. I don't know if it is justified to monopolize the "sequence" page for mathematics; if wikipedia wants to acheive the reputation of a "honorable" reference work, this page must be turned into a disambig page. (And "but there are already so many pages referring to this one" is just one more argument to do this (eventually inevitable) job as soon as possible, the earlier, the better. I'm sure there are enough robots out there who can add " (mathematics)|" to every "sequence" in double-['s.)

   Well, let us see if anybody complains about the monopoly :) If we move this to sequence (mathematics), somebody has to promise upfront to do the disambiguation. It is easy to hope somebody else (or some bot) will do it, but usually these things tend to not get fixed. Oleg Alexandrov 20:40, 8 Apr 2005 (UTC)

2. I think it would be more consistent to define a sequence as a family indexed by natural numbers, rather than to specify {1,2,3,...}. On the former, everybody agrees, on the latter (i.e. 0∈N or not), not - this discussion should not take place here.

   Whatever. :) I thought the {1, 2, 3, } thing was kind of fine. Oleg Alexandrov 20:52, 8 Apr 2005 (UTC)

3. I don't like the intro "list of objects organized in linear manner" - this would include sequences, sorry: nets, sorry: families (?) indexed by real numbers, while it seems to me that sequence does mean, very specifically, that it's indexed by N (maybe translated by some k∈Z).

   I agree with you. But this is for simplicity. You look at things as a serious mathematician. Most people don't. By "linear" most people will understand "beads put on a string", and this is exactly what we would want them to think. Oleg Alexandrov 20:40, 8 Apr 2005 (UTC)

   Concerning both of your previous comments, the problem is that many rather specialized pages from mathematics link to this one, so a "common culture" version of this page is not adequate, IMHO. — MFH: Talk 17:27, 21 Apr 2005 (UTC)

4. In that, I agree with previous posts that "sequence" (in mathematics ;-) should imply "infinite" - a finite family is IMHO called a system, although this must be relaxed for people who call e.g. in quantum mechanics a (Hilbert) basis a complete system. — MFH: Talk 20:18, 8 Apr 2005 (UTC)

It was me arguing for keeping the sequence infinte only. In the meantime, I got to agree with Zero, let us not monopolize sequence for the sake of analysts. The discrete poeple would like to know some sequences are finite. :)

I do not yet agree with this. Maybe "the discrete people" can assume whenever they want that almost all elements are zero, but "sequence" should definitely imply definedness on an infinite subset of N - else most statements in limit (mathematics) (and in many other places) become wrong, I think. — MFH: Talk

Other comments? Oleg Alexandrov 20:40, 8 Apr 2005 (UTC)

OEIS has plenty of finite sequences --Henrygb 23:25, 8 Apr 2005 (UTC)

MFH, you have good and valid points. I will challenge you with one thing though. Try to think very carefully about how this article should look like, so that it pleases both discrete people, and analytic people, and in the same time the article is not too difficult. Basically, try to think how the article should look like without being biased towards either side.

And there are indeed many articles linking here for which the sequence is finite. Here are some examples:

• Pseudorandom number sequence

here the idea is clearly that of an infinite sequence ('repeats from some large number on') — MFH: Talk

• Thread (computer science)

listed in sequence (non-mathematical) — MFH: Talk

• Exact sequence

This truly great mathematical example convinced me: the sequence article needs to be rewritten! (;-) — MFH: Talk

• Molecular phylogeny

broken link, should be DNA sequence — MFH: Talk

• String
• Transparency (telecommunication)

maybe (bits in one packet), maybe not (until connection breakdown) — MFH: Talk

• Sequence motif

not a sequence in mathematics — MFH: Talk 14:56, 26 Apr 2005 (UTC)

Oleg Alexandrov 20:31, 21 Apr 2005 (UTC)

all changes reverted

I am rolling back MFH's changes, which do not have any consensus here and violate normal practice in mathematics. Finite sequences are perfectly ordinary. They are not a "non-strict" usage, and regarding them as infinite sequences with missing terms is ridiculous. --Zero 02:08, 23 Apr 2005 (UTC)

I went to MathSciNet and did a search for "sequence in review text", then looked at the first 20 that came up. In 10 cases the sequences are clearly infinite, in 8 cases clearly finite, and in 2 cases I couldn't tell. In some areas of mathematics (esp. analysis) sequences are usually infinite. In some areas (graph theory, bioinformatics, group theory, operations research), finite sequences far outnumber infinite ones. In number theory and combinatorics, both finite and infinite sequences abound. We can't pass judgment on common practice by claiming that one usage is correct and the other isn't. --Zero 03:15, 23 Apr 2005 (UTC)

Rolling back all of MFH's changes seems a bit drastic. Paul August ☎ 03:36, Apr 23, 2005 (UTC)

Tricky business. I don't know what the right approach is.

Besides the finite/infinte thing, I had a few other issues with what MFH changed. I did not like the recently inserted

Mathematically, a sequence x of objects from a set S is a function, which associates to every element i of the index set I an element x_i of S.

and the i=p, p+1, p+2, ... and the $S_{1}^{n}(x)$ things in several places.

I perfectly understand MFH's motivation. However, making articles more formal/general/rigurous makes them less penetrable for the general public. The encyclopedic format makes it necessary to be less rigurous, skip over details, and even do handwaving. Even with this, many math articles have the reputation of being very hard to understand. Oleg Alexandrov 03:40, 23 Apr 2005 (UTC)

Dear collaborators,

I understand and accept the critics above, but I'm nonetheless sorry that all changes have been undone. Please consider the following:

1. I made an effort to make a very understandable and nevertheless concise introduction. In particular,
   1. I think my formulation of the "linear order" is much easier to understand and at the same time more correct and more universal that the original one ("...comes before, or after, each other element")
   2. I put a "touchable" example of a finite and non-numerical sequence (words of the article), even before the more "usual" example of an infinite, numerical sequence. If you want absolutely to add the "(C,Y,R)" example, why not.
2. I don't understand why the "general introduction" should contain somewhere "lost in the middle" a "more formal definition" which still isn't one, and still isn't general enough to encompass neither the "objects or events" mentioned at the very beginning, neither a sequence starting elswhere than at 1.
   What is the objection against making a cut and start a section with a real and nevertheless general and easy-to-understand "formal definition"?
3. Maybe it was a mistake, but I made many changes at once to avoid you a long list of individual changes. At the same time, I took care to make many subsections to allow selective edits of all I did for those who would not like some particular thing. Indeed, I made many corrections in several places of of the original article that truly needed it, IMHO, without too much touching to the original content.

Summarizing, I would like to insist in that I invested several hours to make what I consider well intended changes, in taking particular care of respecting all of the different opinions, especially

• concerning finite sequences (maybe not good enough, but I could not do better than allowing an arbitrary index set and explicitely discussing finite sequences with the (A,B,C) example in generalizations - maybe should I have put it directly into "Definition" (which I felt a bad idea))
• I didn't delete (almost) anything of what was there - please convince yourself!

I would be glad if Zero0000 would put back my version. Then I would agree to "undo" myself some of my changes which you don't like (e.g. the partial sums in the "Series" section, and the comment that finite = "infinite with finite support", you know what I mean) — MFH: Talk 15:12, 25 Apr 2005 (UTC)

compromise

I admit that my "definition" section was too much focussing on infinite sequences (especially in the oversized "notations" subsection).

But many other things were not better before, especially

• the first phrase
• the series section (a series is NOT the sum of a sequence, but is the sequence of partial sums, which may or may not have a limit, which then is the sum of the series)

So maybe I'll just make some more delicate minor changes...

However, concerning rigour, I think that when a page starts with the (multiple) disclaimers

This is a page about mathematics. For other usages of
"sequence", see: sequence (non-mathematical).
In mathematics, ...

then one should, beyond an introductionary phrase that sufficiently explains the basic ideas and about all that can be said without technicalities, really go to precise definitions. Without precise definitions, a mathematical article is worthless, even worse, it can (and will) induce misunderstandings, confusion, and truly wrong statements in other pages which will be difficult to track.

On the other hand, I think it would be a good thing to start the page with a list of examples like the one given by Oleg above: this would already enlarge the horizon of narrow minded people like me...., and justify the precise formulation of further definitions.

— MFH: Talk 19:23, 25 Apr 2005 (UTC) (posted w/ some hours delay due to connecion problem)

First of all, let us not worry about the

This is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical).

thing. That's meta information and not part of the article.

I agree that a precise definition is needed. But, I thought it was already in the fourth paragraph in the article.

If you would like to talk about sequences starting not at 1 but at something else, that could be maybe mentioned briefly in the fifth paragraph, where bi-infinte sequences are defined. But let us try to avoid the x_i, i=p, p+1, p+2, thing, and rather say that in words (like, a sequence can also start from 0, or from any other integer).

The series paragraph could indeed use work, but again hopefully without introducing new notation. A short blurb and a reference to the series article should be enough.

More examples would be nice. We could see about that later.

And we should not attempt to rewrite this article again. First Zero rewrote it, then I rewrote it, then I rewrote it again, to incorporate Zero's suggestions, then MFH rewrote it. That should be enough of rewriting. :) Oleg Alexandrov 20:21, 25 Apr 2005 (UTC)

I'm glad that the minor modifications I made (before reading your message...) are in agreement with what you write above. (Although I admit that the series "blurb" grew a little too big, and maybe I have to weaken a little what I said about formal power series.)

PS: in fact, I never rewrote the article, but nobody noticed that... — MFH: Talk 15:57, 26 Apr 2005 (UTC)

a little sequential stirring

Ok, lassies and laddies, here's a question: how many sequences (4,2,7,8) are there? You probably think I've lost my marbles, but read on for a moment. Let's restrict ourselves to sequences of integers for simplicity. As our "formal definition" has it, (x1,x2,x3,x4) is a function from {1,2,3,4} to Z (the integers), while (y2,y3,y4,y5) is a function from {2,3,4,5} to Z. These can never be the same function because they have different domains. Yet, when x1=4, x2=2, x3=7, x4=8, y2=4, y3=2, y4=7, y5=8, both these sequences are (4,2,7,8). I'm sure the great majority of mathematicians, while sober, would agree that in these circumstances (x1,x2,x3,x4) and (y2,y3,y4,y5) are both equal to (4,2,7,8) and thus equal to each other. The moral of this story is that we are confusing sequences and indexing functions. I think, after at least 5 minutes of intense concentration, that sequences are not functions at all. Here's what I think they are:

A sequence is a countable multiset with a total order.

So what exactly do we mean when we write (y2,y3,y4,y5)? Well, we are saying there is a function y that maps {2,3,4,5} into integers, and we are choosing the sequence (y(2),y(3),y(4),y(5)). When y has a value, i.e. is a particular function, we get a sequence of integers, like (4,2,7,8). The function y is not part of what the sequence is, it is just the tool we used to construct the sequence. We could construct the same sequence by lots of other routes too. Note that this is different from an array in formal computer science, which keeps the indexing function when it has a value.
Here's another proof that our definition is wrong: (y4,y2,y3,y5), which is perfectly ordinary mathematical notation, is a different sequence from (y2,y3,y4,y5) even when the same function from {2,3,4,5} to Z is used. The problem is that a sequence has an ordering of its own; it doesn't derive its ordering from an indexing function.
Whadyaall think, send Zero to the funny farm? --Zero 09:05, 26 Apr 2005 (UTC)

Zero, now you are going to compete with MFH about who can write the most complicated/correct/all-encompassing defintion over here, right? :) Oleg Alexandrov 15:11, 26 Apr 2005 (UTC)

Dear Zero, finally we agree!

At least on the point that the "formal definition" is not good. In fact, the best, correct, all-encompassing (and least complicated!) definition is given in the first phrase of the article. (Thus preventing the article from the need to be rewritten.) I didn't dare to remove the "formal definition", but it has now slipped into the "Examples and notations" section.

Your "ordered multset" idea is not good, because the order is not on the elements, but on their indices.

No, the ordering is on the elements. The sequence (apple, banana, orange) is the multiset {apple, banana, orange} with the total order "apple < banana < orange", or in ordinary English "apple is first, then comes banana, and orange is last". There is no need for indices. --Zero 23:52, 26 Apr 2005 (UTC)

But a function on {2,3,4,5} isn't a sequence (according to the "formal definition"), so there is no problem of ambiguity.

I still think that the best definition (distinguishing a sequence from a net with arbitrary index set) is that of a map defined on some subset of Z, and for a sequence in the "strict sense" (say), the subset should be bounded from below, and if the sequence is infinite, contain all integers large enough.

— MFH: Talk 15:35, 26 Apr 2005 (UTC)

I indeed think the series thing is a bit too long. I saw you plan to modify some things in the series article. After you do that, could you maybe remove some of the long explanations you put here? I would think this article should only talk about the connection between series and sequences, more than that could go a bit offtopic. Oleg Alexandrov 17:57, 26 Apr 2005 (UTC)

I agree. the "Series" part was too long even before... — MFH: Talk 20:00, 29 Apr 2005 (UTC)

Notation

Hi, under "Examples and notation", could you also include how to denote "belongs to", i.e, x belongs to sequence S. what's the symbol? is it the same as the set membership epsilon symbol? thanks. --anon

I never encountered notation for saying that an element is in a sequence. Well, if you really wish, you can write for the sequence {a_n}

$\exists n\ a_{n}=x$

but that I guess is just a way of saying in symbols that one term a_n equals x, and is not a specialized notation. Oleg Alexandrov 15:25, 13 August 2005 (UTC)

I don't think there's any special notation. I would use the set membership symbol if the intent was clear from the context. --Zero 01:46, 14 August 2005 (UTC)

Terms of a series

Hi. I think in the last paragraph Sequence#Series, in the phrase "This new sequence is called a series with the terms x1,x2,x3,... and is denoted" the terms are instead s₁, s₂, s₃. What do you think?
Stefano85 23:58, 23 January 2006 (UTC)

Not sure, but I think the terms of the sequence are still x1, x2, .. etc. The s1, s2, ... are the partial sums, but not the terms. The difference between a sequence and a series is that in a series the terms are added up. But again, not perfectly sure. Oleg Alexandrov (talk) 03:01, 24 January 2006 (UTC)

I, too, had been somewhat uncertain about what the terms of the series are, as indicated by my previous edits and current reversion, with some additional clarification. Through some discussion with colleagues, I have come to understand that the terms of a series are indeed the x's, not the S's. Without delving fully into it, I have made the entry here consistent with the statement under the entry for "series" that the formal definition of a series is that it is the pair of sequences of x's and S's, with the x's comprising the terms. I welcome any further clarification. KCliffer 20:35, 30 September 2006 (UTC)

Range of integer values

In for example computer science and numerical computing, the range of values, or the sequence (of integer values), from a to b refers to a, a+1, … b. Is this okay to mention in this article? Or what mathematical terminology (in words) is appropriate for this? Can "series" be used somehow? Mange01 23:49, 3 December 2006 (UTC)

Solving for the general term

Where can I find rules for solving for the general term? For example, if I define something like $a_{1}=1,a_{n+1}=q\ a_{n}+n!+\sin(a_{n})\,$, there's a great chance that I can't find a general formula for a_n, but for simpler cases like $a_{1}=x,a_{n+1}=m\ a_{n}+p\,$, I may have a solution (like setting $b_{n}=a_{n}+k\,$ and forcing b_n to be a geometric progression). Albmont 13:41, 3 January 2007 (UTC)

I added a link to Recurrence relation.--Patrick 09:51, 2 June 2007 (UTC)

Countable? Why?

There is nothing about a sequence that suggests that it should be a function from a COUNTABLE set. It simply needs to be a function from a TOTALLY ORDERED set.

For example, imagine a Poisson random process (or really any Continuous-time Markov process or counting process in general). This is a SEQUENCE of random variables that is OFTEN indexed by a set of nonnegative real numbers (e.g., representing time).

Additionally, I think that this definition could be made more general with a more set theoretic definition (filter bases and all that). --TedPavlic 17:15, 10 February 2007 (UTC)

We follow the existing mathematics terminology. A sequence is per agreed conventions in mathematics indexed by a subset of the natural numbers (or integers).

I will very strongly disagree with making things more general. Wikipedia is a general purpose encyclopedia. Articles should be approachable as much as possible by the general public, especially introductions. Oleg Alexandrov (talk) 18:41, 10 February 2007 (UTC)

I would say (though I lack a source) sequences with real indexes, instead of natural numbers, are no longer called sequences. While it is possible, and often needed, that elements of an array depend on a real number (sometimes even not converging towards infinity, but limited inside a certain range), this is actually expanding of the term sequence, but is not a sequence per se. --77.46.217.150 (talk) 19:30, 25 September 2009 (UTC)

I suspect there is confusion here between indexing and relative order. Something like (x_α, x_β, x_γ, ... ), where α, β, γ, .. are real numbers (or complex numbers, or values from some weird esoteric space) is a perfectly ordinary sequence that already fits the definition in the article. The mapping from {1,2,3,...} required for the definition has nothing to do with the naming scheme of the elements, it just specifies what is first, what is second, etc. The mapping is 1→x_α, 2→x_β, 3→x_γ, etc.. The other mapping visible in the example, namely α→x_α, β→x_β, ..., comes from the application domain and is not relevant here. Mind you, I have seen something like (x_α) called a "sequence" when α ranges over all real numbers, but I think that is an old usage which is now considered quaint. Adding it to the article (especially near the front) would make the article less useful, imho. Zero^talk 02:09, 26 September 2009 (UTC)

Sequence

What are the differences between the mathematical concepts of sequence and interval? They should at least reference each other, even if only in "see also". -unsigned

A sequence is countable, while an interval is continuous, for example. Oleg Alexandrov (talk) 16:24, 17 May 2007 (UTC)

Digital video editing

I removed a new section on this since it doesn't belong in this article, which is about mathematical sequences. It could, of course, go in a more suitable article, but this one relates to mathematics. Xantharius (talk) 16:49, 19 March 2008 (UTC)

Is a sequence a type of set?

The second sentence's first 3 words are "like a set" which means no.

My second question, is can someone who is familiar with all three topics check "sequence" "set" and "partially ordered set" for style consistency? I have read all three, and its congruency is a little bit confusing. There are no contradictions, but as I'm learning this from wikipedia and not by a textbook (who the same author sometimes writes related chapters and the consistency is benfecial to the reader) I can sense some differences which maybe this article could stand benefit by adopting. This page reads like it was definitely master-edited by an PhD while http://en.wikipedia.org/wiki/Partially_ordered_set is written by someone who focuses on educating the user (making no sacrifices in quality of the article), even including an example immediately after the introductory paragraph. Sentriclecub (talk) 01:03, 28 May 2008 (UTC)

Well, almost everything in mathematics is a set. A sequence of elements from a set S is a function from the natural numbers N into S, and is therefore a subset of N × S, but this is hardly a very helpful way of thinking about sequences. The writing of this article seems fairly informal, but could do with editing in both directions (more informal examples for those who want them; more formal definitions for those who need to know exactly what a sequence is). Maybe it's time to do both to this article. Xantharius (talk) 22:17, 28 May 2008 (UTC)

indexing of a sequence, revisited

We define a sequence as a mapping from {1,2,...} onto S. OK. But then we say "Sequences may also start from 0, so the first term in the sequence is then a₀." This seems to be inadequate (as well as badly worded). Consider very common cases like (a₂,a₄,a₆,...). I think the problem is that the text confuses two different things. The mapping from {1,2,...} onto S indicates the position of each element in the list; it doesn't have anything to do with the names of the elements. That is, (a₂,a₄,a₆,...) means the mapping 1→a₂, 2→a₄, 3→a₆, ... . I propose to fix this as follows (to replace paras 2-4 of the "Examples and notation" section; please comment.

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

For example, the sequence of prime numbers (2,3,5,7, … ) is the function 1→2, 2→3, 3→5, 4→7 … .

In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names convenient for the application. For example a sequence might be written as (a₁, a₂, a₂, … ), or (b₀, b₁, b₂, … ), or (c₀, c₂, c₄, … ),

Zero^talk 01:23, 18 September 2009 (UTC)

I think the "without zero" is quite irrelevant and somewhat misleading. Yaan (talk) 12:25, 21 September 2009 (UTC)

The problem is that some people include 0 in "the natural numbers". But you are right, it is clear enough to write {1,2,3,...} without the parenthetical comment. I struck it. Zero^talk 14:13, 21 September 2009 (UTC)

definition?

I think it would be a good idea to make a definition of what a series is easier to find. In fact, I personally believe a definition is more important than all the examples and other stuff currently given. A sequence is not some arbitrary poset, it can not be of uncountable length etc. Yaan (talk) 12:13, 21 September 2009 (UTC)

Cauchy sequence in R - not convergent?

I see this illustration on the top of the page, which is named (i.e. the file itself) "Cauchy sequence illustration 2", and followed by text: Infinite sequence of real numbers [which is] ... neither ..., nor convergent. This must be some kind of blunt mistake, because in R (real numbers) all Cauchy sequences are convergent. Maybe the displayed sequence is really not convergent, but in that case the picture's name is misleading and it can't be a Cauchy sequence - Cauchy sequences of real numbers are always convergent. --Дарко Максимовић (talk) 01:45, 24 September 2009 (UTC)

The sequence in question is not a Cauchy sequence. The title of the image is not right, but we generally don't worry about titles of images. The caption did not claim the sequence was Cauchy, but I added "nor Cauchy" to it. — Carl (CBM · talk) 01:48, 24 September 2009 (UTC)

Thank you. It was somewhat disleading, for I, for myself, for a moment thought it was trying to say it was Cauchy. Obviously, some of us do take a look at a picture's title, so it's better this way. --77.46.217.150 (talk) 19:25, 25 September 2009 (UTC)

Why should finite/infinite link to finite/infinite set?

I removed the links because "finite" here means finite sequence, not finite set. someone reverted my good faith edits. please explain your rationale here. Pagen HD (talk) 14:51, 25 February 2012 (UTC)

A finite sequence, in set theory, is also a finite set; either as an ordered "tuple", or as a function with domain a natural number. An "infinite sequence", in set theory, is a function with domain N, which is also an infinite set. — Arthur Rubin (talk) 15:56, 25 February 2012 (UTC)

exactly the same elements can appear multiple times at different positions in the sequence ?

I don't understand this sentence at the beginning of the article.

What kind of order can have different truth values on the same couple of objects ? For instance, if (x, y, x) is a sequence (x being the same element appearing multiple times at different positions), by definition of total order (the definition does not make sense in case of partial order) that means that x ≤ y and y ≤ x ? Then x = y ? An I missing something ?

An answer

The article has certainly changed a lot since you posted this question, but I believe the answer to your question is that the elements in the sequence are distinguished completely from each other, although they can have the same "sequence" value. One definition of a sequence is a map from a countable totally ordered set to some space. In terms of set theory, this means that the sequence is a set of ordered pairs $\{n,a_{n}\}$. Therefore, the sequence {x,y,x}, written as a set would be $\left\{\{1,x\},\{2,y\},\{3,x\}\right\}$. Brent Perreault (talk) 02:07, 20 December 2012 (UTC)

Did an extensive revision

It seemed that this article was due for a makeover, and had fallen behind in quality (and coverage) compared to similar articles such as series (mathematics). For this reason, I reorganized the article and expanded its coverage approximately two-fold. My focus here was to clarify technical discussion, and to lengthen the introductory parts and conceptual discussion. Moreover, I added some section I deemed extremely important, such as a section on convergence and a subsection on bounded sequences.

These changes were carefully planned and carefully written in a manner appropriate for an article that receives over 1000 views per day. I request that any large objections to the new material be carefully made by changing the parts of the article that are deemed inappropriate or poorly presented. Or, by bringing up those issues in this forum. I am asking that other editors refrain from reverting all of my changes so that we can continue to improve this article the most efficient way possible. Thanks,

Brent Perreault (talk) 02:20, 20 December 2012 (UTC)

Possible Typo?

Hello. Is the "\" in this sentence from the article a typo?: "For example, {M, A, R, Y\} is a {sequence} of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}." Thank you to the editors of this article. 68.196.183.70 (talk) 19:12, 29 December 2012 (UTC)

Yeah, typo fixed. Thank you! Feel free to delete this post now (delete the subsection 'Possible Typo', that is). Brent Perreault (talk) 21:25, 29 December 2012 (UTC)

Ambiguous bi-infinite sequences

Should bi-infinite sequences be rendered like $(\dots ,-4,-2,0,2,4,6,8,\dots )$ ? After all, that notation is ambiguous (consider $(2z+2)_{z\in \mathbb {Z} }$, which could be rendered the same way). I don't know of any better notation, so should we just quickly point out that ambiguousity and that bi-infinite sequences should always be written as families if actually used? 78.51.147.63 (talk) 09:33, 19 April 2011 (UTC)

Different ways to index the "same" sequence is a problem of listing sequence elements in general. For instance, the sequence {1,2,3,...} can be "rendered" in multiple ways. In the case where the indexing set is assumed to be the natural numbers, then the ambiguity is no longer an issue, so it is less of a problem in that case. However, in general, sequences are sometimes considered "the same" if they give the same list of ordered elements (independent of indexing) while other times the indexing is considered central to the specification of the sequence. Thus, the issue is of convention, and while the standard in real analysis is to have the specification make two sequences different, in listing important sequences such as the Fibonacci numbers the numbering matters very little. Brent Perreault (talk) 21:34, 29 December 2012 (UTC)

List of sequences/ Types of sequences

Does wikipedia need a list of sequences or types of sequences page? Brad7777 (talk) 22:11, 8 November 2011 (UTC)

Sure! Brent Perreault (talk) 21:35, 29 December 2012 (UTC)

Multiplicative?

I've never heard of a multiplicative sequence. A multiplicative function is what is described, even if it is technically a "sequence". — Arthur Rubin (talk) 09:23, 11 December 2010 (UTC)

Brent Perreault (talk) 21:54, 29 December 2012 (UTC) I checked the reference and the way it is now written is correct (exactly as found in the book with the a_n index notation). However, there are a number of other uses of the term and it is not clear to me which are popular. In any case, all the common uses that I could find are listed in encyclopedic form. Brent Perreault (talk) 21:54, 29 December 2012 (UTC)

Archived Posts before 2010

Archives help us keep this page clean and quick to load, allowing users to quickly see current information. For older posts see the archive linked at the top (right) of this page. Brent Perreault (talk) 22:18, 29 December 2012 (UTC)

braces or parentheses?

I came to this article looking for some confirmation that a convention I have always used is a "standard" convention:

braces {M,A,R,Y} are used to denote (unordered) sets

parentheses (M,A,R,Y) are used to denote (ordered) sequences

I notice that the article uses braces for sequences. I am still unsure this is "standard" so I won't edit... but thought I should raise it

Agneau (talk) 15:05, 14 February 2013 (UTC)

In some contexts, where unordered sets are rare, braces are sometimes used for sequences. I believe you are correct, except that $\{a_{n}\}_{1}^{\infty }$ is often used instead of $(a_{n})_{1}^{\infty }$ as a compact notation for sequences, and sometimes a different symbol is used for multi-sets. I would bring it up on WT:MATH, but I would have no objection to changing most of the braces to parentheses. — Arthur Rubin (talk) 15:20, 14 February 2013 (UTC)

• As far as I know, this article used round brackets for sequences up until #Did an extensive revision mentioned above, so there may be consensus to go back. Also, I suspect using different notations would help distinguish sets and alphabets from sequences. Vadmium (talk, contribs) 23:46, 14 February 2013 (UTC).

• Just looking in books within reach... Serge Lang uses braces in his Analysis. Briggs and Cochran (boo) also use braces. Having done nothing but set theory lately, I felt sequences should look like $\left(a,b,\ldots \right)$ or $\langle {}a,b,\ldots \rangle {}$, but I guess it's not an issue in fields that don't talk a lot about ZF sets. melikamp (talk) 03:56, 8 March 2013 (UTC)

OK, just now I changed the braces back to parentheses. Actually, angle brackets might be the most suitable of all for my taste and experience, which is limited but not too limited. But this way is fine too. I'll leave them as parentheses because they were that way before December 20, 2012. --Hoziron (talk) 02:36, 16 April 2013 (UTC)

Looks good. I believe this notational issue should be part of the article. I would bring it up in the first section, perhaps the second paragraph. If there are no objections I will change insert it myself. Brent Perreault (talk) 21:57, 3 September 2013 (UTC)

Please do. But don't forget to mention that frequently no brackets of any sort are used. Perhaps the most common notation for a sequence is just a list with commas: a, b, c, ... . McKay (talk) 05:20, 4 September 2013 (UTC)

Limit of a sequence

Why does this article only define convergence of a sequence in a metric space, rather than a general topological space? I know there's a link to "Limit of a sequence," but at the very least this page should give the reader some indication that there's a major concept missing here, and that they need to follow that link to see it. — Preceding unsigned comment added by 165.123.213.202 (talk) 08:09, 10 December 2013 (UTC)

I agree on this. Martinkunev (talk) 19:20, 27 January 2014 (UTC)

Phrase (one-sided)

There are three occurrences of the phrase "(one-sided) sequences", beginning with the section "Definition of convergence". Please define the term. (reader whose mathematical level doesn't quite reach that of the writers) Mathyeti (talk) 17:02, 12 April 2015 (UTC)

Probably, it is used to distinguish from doubly infinite lists, introduced in section Sequence#Finite and infinite. I suggest to introduce the term there. - Jochen Burghardt (talk) 19:52, 12 April 2015 (UTC)

I copy-edited that section, to define the notion of one-sidedness explicitly. - Jochen Burghardt (talk) 21:58, 15 April 2015 (UTC)

Define "list"

This article gets off to a curious start by saying "A sequence is an ordered list", and list is a disambiguation page. What exactly is the definition of "list" in a mathematical context? Horatio (talk) 01:06, 3 May 2014 (UTC)

Maybe it's just used as an English word, and not a technical term? In that case, it should be unlinked, so that the reader isn't tempted to follow it to the disambiguation page. Horatio (talk) 01:14, 3 May 2014 (UTC)

I guess I'll unlink it. By all means relink it to an appropriate technical article, if relevant. Horatio (talk) 22:43, 4 May 2014 (UTC)

A list is a collection of objects whose orders are important(unlike set and multiset) and repetitions are allowed(unlike set). Therefore the word "ordered" in "ordered list" is redundant. It should be just "list". Any objections? LoMaPh (talk) 00:54, 22 February 2015 (UTC)

Your definition is common among computer scientists, but (unlike that of a mathematical set) hardly known to any other people. Therefore, while your redundancy argument is right, "..., a sequence is a list" doesn't explain anything at all to most people. But what about "..., a sequence, a.k.a. list, is a collection of objects whose orders are important and repetitions are allowed." Would that be ok? - Jochen Burghardt (talk) 06:29, 22 February 2015 (UTC)

Sounds good. LoMaPh (talk) 02:03, 25 February 2015 (UTC)

Why do you say "... whose orders are important ..." rather than "whose order is important"? Mathyeti (talk) 17:05, 12 April 2015 (UTC)

You are right. - If there are no objections to the accordingly corrected version, I'd insert it in the article's lead. - Jochen Burghardt (talk) 19:52, 12 April 2015 (UTC)

Attempting to perform the insertion, I found that D.Lazard did it already on 25 Feb. - Jochen Burghardt (talk) 22:00, 15 April 2015 (UTC)

Thanks all, the current version is much better. Horatio (talk) 01:56, 31 July 2015 (UTC)

Shadow sequence

Perhaps, one can explain for God's sake what a shadow sequence is. I couldn't find an easy example anywhere. (They are also used to define annihilating prime numbers.)--Sae1962 (talk) 13:39, 8 February 2016 (UTC)

Article should be restructured

Currently, wide parts of the article apply only to (cum grano salis) number sequences, while the definition admits arbitrary mathematical objects to occur in a sequence. Imho, the article should be restructured into sections handling different kinds of objects, and presenting only properties that make sense within its section. For example,

• "Sequence#Cauchy sequences" should be moved below a section named "Sequences in metric spaces";
• "Sequence#Increasing and decreasing" should be moved below "Sequences of real numbers", or maybe, "Sequences in totally ordered sets";
• "Sequence#Series" may belong below "Sequences in vector spaces", or "Sequences in groups" (or even magmas?).

An introductory section should discuss the relations between different kinds of target sets, and also relate the most common applications to these kinds (saying e.g. "Since the set of real numbers is totally ordered, a metric space, and a vector space, the concepts of Cauchy sequences, of increasing and decreasing sequences, and of series can be defined there.")

As another issue, sets of sequences should be handled together in an own section, discussing there Sequence#Sequence spaces as well as Sequence#Free monoid, and mentioning string (computer science), formal language, and ω-language there. - Jochen Burghardt (talk) 11:17, 18 April 2016 (UTC)

Definition of sequence

Hi, I am changing the definition of "sequence" in the article and I want to describe my motivation.  First of all, the original text said that a sequence is "usually defined" as a function whose domain is a countable totally ordered set, but no references are given for this.  I would be surprised if any exist, since this is a very strange definition of a sequence.  This leads me to my second point, which is that this definition includes as sequences functions whose domains are the set of rational numbers or similarly crazy countable sets.  In standard mathematical practice these are not usually considered sequences.

Now I think I understand the motivation behind the previous definition of sequences, which was to allow finite and bi-infinite sequences to be counted as sequences.  I've come up with a new definition that includes these as sequences while excluding things like functions whose domain is the rationals.  I've also reworded to make it clear that this is not a standard definition, just a convenient one.

I've also reworded the citations to avoid giving the impression that there is broad consensus on how to define a sequence within any given field.  As far as I know this is not true, although I would guess that the most common definition (in all fields) is that a sequence is any function whose domain is N.

David9550 (talk) 15:33, 21 January 2016 (UTC)

I agree that the first definition given for sequence includes as sequences functions whose domains are the set of rational numbers or other totally ordered sets.  While the second definition, beginning with the words "Formally, a sequence can be defined as a function whose domain .." corrects the initial definition, I do not agree that the initial definition should be left incorrect.  I believe totally ordered is inadequate to describe a sequence.  I suggest that well ordered would be a closer description for a sequence.  Also the type of collection to which this well ordering is to be given is reasonably well known as a multiset, so why not use this in the initial definition?

collections

	no duplicates	duplicates allowed
unordered	set	multiset
partially ordered	poset
partially ordered with meets and joins	lattice
totally ordered	chain
well ordered	chain sequence	sequence

Including the above table at the beginning of the article might be helpful.  I wish I knew names for the missing entries.

So, as a recommendation, I suggest changing the first sentence to "a sequence is a well ordered multiset."  This change will allow deletion of the sentences contrasting set with multiset and ordered with unordered, necessary only because the ambiguous term collection was used instead of the required term multiset.

When it becomes desirable to introduce double ended infinite sequences, one could refine the concept of well ordered, distinguishing collections that have a first element (well ordered) from those (sequentially ordered but not well ordered) that have both a next and a prior member for each member.

Howard McCay (talk) 05:53, 4 June 2016 (UTC)

The term "well ordered multi set" is too WP:TECHNICAL for the lead: the reader of this article is not supposed to know what are a well order and a multi set, which are less elementary than the concept of a sequence.  Moreover the term is incorrect, as transfinite ordinals are not used for indexing sequences.  However, I agree that "total order" is also incorrect and possibly misleading.  Therefore, I'll replace "ordered" by "enumerated".  The advantage of using such a non-technical term is not only to avoid confusion (if not well understood, the reader must go to the formal definition), but also to follow the recommendation of MOS:MATH:

The lead section should include an informal introduction to the topic, without rigor, suitable for a general audience.  (The appropriate audience for the overview will vary by article, but it should be as basic as reasonable.)  The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal and correct approach.

D.Lazard (talk) 06:58, 4 June 2016 (UTC)

Thank you for the clarification.  I had forgotten that well ordered collections can be transfinite (if you accept the axiom of choice).  I agree that these won't do as sequences (unless you wish to allow transfinite sequences).  You are also correct in asserting the principle of non-technicality in the lead.  So I like your use of the commonly understood term enumerated.  I do not believe that multiset is a less elementary concept than sequence.  However, you are correct in that multiset is a less familiar term than sequence.  I added internal links for the terms enumerated and collection.  It is a shame that I could find no article explaining these terms from a mathematical viewpoint, only articles from computer science.  The problem with using a computer science definition for a mathematical term is that computer science is concerned also with implementation and not merely with the mathematical concept.  Howard McCay (talk) 15:38, 4 June 2016 (UTC)

It is not a surprise that these terms are not explained in math articles: they have no mathematical meaning, except their dictionary definitions and Wikipedia is not a dictionary. For emphasizing that these terms must be taken in their common, informal sense, I have unlinked them, and added a {{wiktionary}} template that links to them. D.Lazard (talk) 16:43, 4 June 2016 (UTC)

Ban Numbers

The citation of ban numbers as an interesting sequence seems random and a distraction to the article at large. It certainly doesn't qualify as an important example. I suggest it be removed as unnecessary noise. — Preceding unsigned comment added by G a adams (talk • contribs) 12:43, 10 July 2016 (UTC)

Removed the example and all occurrences of "important" in the section. D.Lazard (talk) 13:23, 10 July 2016 (UTC)

Misleading statement

"Many authors also impose a requirement on the codomain of a function before calling it a sequence, by requiring it to be the set R of real numbers,[2] the set C of complex numbers,[3] or a topological space.[4]"

This is not true. None of those authors would deny the validity of other sorts of sequences. All they are doing is telling us what sort of sequence they are going to tell us about. McKay (talk) 04:33, 23 April 2018 (UTC)

I Agree. — Carl (CBM · talk) 16:42, 23 April 2018 (UTC)