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WikiProject Mathematics (Rated C-class, Top-importance)
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One of the 500 most frequently viewed mathematics articles.


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 an 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)

Ambiguous bi-infinite sequences[edit]

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? (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[edit]

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)

Why should finite/infinite link to finite/infinite set?[edit]

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 ?[edit]

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[edit]

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[edit]

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?[edit]

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. (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)

Archived Posts before 2010[edit]

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?[edit]

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[edit]

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 (talk) 08:09, 10 December 2013 (UTC)

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

Define "list"[edit]

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)


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)