Talk:Sequence

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One of the 500 most frequently viewed mathematics articles.

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)

Contents

[edit] 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 (ai) is a shorthand for the sequence (a1,a2, ...). If it is desired to indicate the range of values i takes, notations such as (ai)5i=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)

[edit] 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:

here the idea is clearly that of an infinite sequence ('repeats from some large number on') — MFH: Talk
listed in sequence (non-mathematical) — MFH: Talk
This truly great mathematical example convinced me: the sequence article needs to be rewritten! (;-) — MFH: Talk
broken link, should be DNA sequence — MFH: Talk
maybe (bits in one packet), maybe not (until connection breakdown) — MFH: Talk
not a sequence in mathematics — MFH: Talk 14:56, 26 Apr 2005 (UTC)

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

[edit] 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 xi 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)