Talk:Limit of a sequence

Really like this page
The intro really helped me understand the definition since in my book it states that for some n>=N dependent on epsilon if d(p sub n, p) 0, then p is the limit of {s sub n}. Really unclear. Thanks wikipedia. Sorry if I put this comment in the wrong place.

Here is an exercise from my book: 1) Prove if {s sub n} converges then {|s sub n|} converges. Is the converse true?

No, if you consider the sequence sn=1 if n is even and sn=&minus;1 if n is odd, then the absolute value has a limit but not the original sequence. Thenub314 (talk) 13:59, 17 October 2008 (UTC)

So the converse isn't true, but what about the proof part? —Preceding unsigned comment added by 144.92.23.234 (talk) 19:06, 21 October 2008 (UTC)

Exercises.
Some exercises, resolution of exercises and stuff like that could be interesting... and helpful for college homeworks :)

Wikipedia is an encyclopaedia, not a textbook. There are probably a lot of textbooks in your college library full of exercises with solutions, or you could try WikiBooks - http://en.wikibooks.org/wiki/Calculus/Limits/Exercises Unnachamois (talk) 17:53, 5 April 2013 (UTC)

How?
ok, i'm still a little confused. How do you take the limit of a sequence? when you do a function you more or less plug it in. but in a sequence what do you do? for instance, if the sequence is (n-1)/n...i can loically look at it and say "well, it will irst be 1/2, then 2/3, 3/4, 4/5, 5/6...etc. which increasing but never getting to one. so the limit is 1. but is there now "way" to do it as with the function? is it merely thinking about it logically?--Jaysscholar 09:56, 4 October 2005 (UTC) (yes, i am sort of asking for help of my hw, but i also think the article should be expanded to include that stuff.)

using merely logic can throw u off. for instance (2n+1)2/(3n-1)2 converges to 4/9. why? the bottom is getting bigger faster so shouldnt it go to 0? Im sure my logic is wring but some one sould make it clear in the article and then tell me so i wont be confused.--Jaysscholar 10:08, 4 October 2005 (UTC)


 * Excuse the unprettiness of the following, I don't know how to format this to make it look pretty, but anyhow... I think the article should have included that limit as n goes to infinity of (1/np)=0 for p>0. That property is very key for finding the limit of a sequence when the sequence can be written as polynomials.  The article maybe should also include the obvious property that the limit of a sequence that is constant is that constant. i.e. let c be any constant, then the limit as n goes to infinity (or goes any number for that matter) of c is c.  The basic idea is to get rid of any variable raised to a power, and leave yourself with just constant terms along with terms of the form 1/np p>0. In your first example above if you take (n-1)/n and multiply by (1/n)/(1/n), which is the same as mutliply by one so as not to change the expression, you get (1-1/n)/(1).  From this form you can use the division and addition properties listed in the article to say that Lim [(1-1/n)/1] = Lim (1-1/n) / Lim(1) = [Lim (1) - Lim (1/n)] / [Lim (1)] = (1-0)/1 =1.  The second example you gave would be very similiar, but first you would have to rewrite the numerator and denominator through expansion, and then multiply the expression by (1/n2)/(1/n2).  If you do this you'll see that it converges to 4/9.--EK711 06:09, 6 October 2005 (UTC)


 * Here's a shorter explanation: n^2 grows so much faster than n or any constant that you can just ignore those terms. Then the function basically looks like (4n^2)/(9n^2) in the long run, which is clearly = 4/9. 198.59.188.232 20:59, 6 April 2006 (UTC)


 * When you "do" a function, you DON'T plug in the value. The definition of the limit of a function at a point x is independent of the value of the function at x, if the function is defined at x at all. As for finding the limit of a sequence given in closed-form, one can use some standard limits like those on the talk page below, and some inequalities (for example if one convergent sequence is less than or equal to another convergent sequence term-by-term, then the limit of the first is less than or equal to the limit of the second). --Kprateek88 ( Talk |  Contribs ) 14:34, 29 October 2006 (UTC)

Basic Examples to help calculate limits more readily?
The more I think on it the more I'm unsure if these should be considered properties of limits, but I wouldn't mind seeing a couple of other simple examples being thrown into the article to help people calculate limits of sequences more readily. Namely:


 * The limit as n goes to infinity of 1/np=0 for p>0.
 * The limit as n goes to infinity of an=0 if |a|<1.
 * The limit as n goes to infinity of n(1/n)=1.
 * The limit as n goes to infinity of a(1/n)=1 for a>0.

I would try to edit it myself, but I'm still pretty new here at the Wikipedia and wouldn't be able to make it look nice. At least they can be here for anyone that may need them for now.--EK711 06:31, 6 October 2005 (UTC)

I might add a couple of these, but first I want to point out that in the last one, your restriction isn't required. Since it's a limit to infinity, nothing that happensless than zero affects that limit. In fact, restriction of that form are never required in limits, (to the best of my knowledge), because those restricted valuess either change the limit, which simply changes the limit, of do not contain the limiting point, in which case they do not affect the limit. He Who Is 22:43, 25 May 2006 (UTC) (Nevermind. I sisn't realize the restrictions were on the constants, not the variable being limited.)

Cesaro Limit
It would be nice to add also the notion of cesaro limit.

Wikipedia sucks for math
Topological space? Limits of sequences are taught to beginning (FIRST YEAR) calculus students! What are you doing including this here? -Iopq 14:23, 28 March 2006 (UTC)


 * It is necessary to have a fully general definition, as this page is not just for first year calculus students. I have added a comment that two of the definitions are the same as the usual one for sequences of real or complex numbers, for those who wouldn't find this obvious. Elroch 18:43, 28 March 2006 (UTC)


 * Everyone knows what a limit is after their first semester of calculus. This page IS just for first year calculus students. Or maybe high school students that are curious. Give me a calculus textbook that doesn't cover limits. YOU CAN'T, THERE ISN'T ONE. Nobody who knows what a topological space needs to know what a limit is. And if they do for some reason like memory loss, then a first-year calculus definition is sufficient. -Iopq 00:26, 23 May 2006 (UTC)

You realize that essentially what you're saying is that this article is useless because there are other ways to attain the information. How does that logic work? And as for first-year college students and curious highschool students, two problems. First of all, Calculus is available at most (U.S.) high schools. Secondly, I'm in middle school. Does that mean I have no interest in anything above the information fed to me by a condescending educational system so clearly based in it's creators' fascist desire to console their own inferiority complex? Wikipedia is a source of knowledge. To remove some of that knowledge because Wikipedia is not the only source of knowledge seems to me to be somewhat counter-intuitive considering that your statement advocates a formal educational system, which is arguably one of the least effective methods of distributing information. He Who Is 22:36, 25 May 2006 (UTC)

The epsilon definition of a limit for me at least didn't come until Analysis in my 2nd or 3rd year of college. Definitions are crucial. And the topological definition is very useful to, perhaps, a student of calculus who knows a regular limit, but not a generalized limit. —Preceding unsigned comment added by 74.66.240.51 (talk) 14:13, 22 May 2008 (UTC)

Just wanted to mention that, as someone who has studied analysis but not much topology, I found the definition of a limit on a topological space very helpful. Rckrone (talk) 07:34, 19 June 2009 (UTC)

Maths is a subject that doesn't lend itself well to a "one size fits all" article about a specific subject, as inevitably it is too complex for some and too simple for others. For most articles, Wikipedia can be a both a learning resource and a reference resource, but unfortunately, when it comes to maths, it is quite hard to please learners at different levels, as what is obvious to some is too advanced for others. But whatever level you're at, there are loads of resources out there to help you learn - just not necessarily Wikipedia, which is for reference as opposed to learning. Unnachamois (talk) 18:00, 5 April 2013 (UTC)

Encyclopedia should be accessible to laymen, this article is not.
Encyclopedia should be accessible to laymen, this article is not.

If I hadn't had a solid secondary school education, I would not understand a word of this explanation. This is not the place for specialized lingo, this is the free encyclopedia that should help especially those who were not blessed with, or could not afford to build, a strong math background.

Please re-write for the mainstream.

Umberto Torresan

''I agree it should be worded more for the layperson, though it begs the question of whether all technical articles should follow this format. It is rather tedious to go through the basics sometimes.''

Piepants 02:55, 8 April 2006 (UTC)Piepants


 * In response to reader requirements, I have added an intuitive description of what a limit of a sequence is at the start of the article. I have also simplified and re-ordered the formal definition section in a way which better complies with wikipedia guidance on the structure of mathematical articles, and which should be more accessible. Elroch 23:09, 10 April 2006 (UTC)

Infinitely Small
Consider the following:

$$0.9999999999... = N$$

$$9.9999999999... = 10N$$

$$9 = 9N$$

$$N = \frac{9}{9}$$

$$N = 1$$

0.999999999... = 1, which someone looking at the limiting value of it's expansion can find easily. And yet it is clear that it is i fact less than one. The largest possible number less than 1. Therefore 1 - 0.999... Is the smallest number greater that zero, although its decimal expansion converges to zero. Therefore, arguably, there is no way to find the exact limiting value of any sequence by analyzing its members, since that technique could give that a sequence limited by 5 could pe limited by 5, 6-0.999..., or 4+0.999..., whose values are also limited by five. Anyone wish to comment on the subject?He Who Is 22:52, 22 May 2006 (UTC)


 * There is no real number which is the " largest possible number less than 1", since for any real number x less than 1, x + (1-x)/2 is greater than x but less than 1. Paul August &#9742; 23:18, 22 May 2006 (UTC)

And when x = 1? Then the ouput is 1, which is neigher greater than x, nor less than one. Also, your claim that there is no largest possible number less than one, is untrue. See Infintesimal. He Who Is 19:37, 25 May 2006 (UTC)

Infinitesimals are not real numbers. See non-standard analysis and hyperreals. &mdash;Tobias Bergemann 07:36, 26 May 2006 (UTC)

Ah. sorry. I missed the "real" part and thought you just said number. He Who Is 11:11, 26 May 2006 (UTC)

Sequences with finite elements
Does “limit of a sequence” have any meaning when the sequence in question has only a finite number of elements? Morris K. 02:21, 27 July 2007 (UTC)

p > 0, or p > 1?
Does anyone have a categorical source, or at least a proof, on the lower bound for p, in the example in the article? I've gone back and forth between the two bounds, but can't decide on anything. This talk page has p > 0, and some edits have changed it to p > 1. — metaprimer (talk) 05:52, 10 November 2007 (UTC)

Informal language?
I understand that limit is very basic concept, but isn't it better to reference an aricle about formal language rather than let unprepared reader get sunk in those lengthy informal "descriptions"? Also, where is Cauchy criterion? Lex aver (talk) 09:55, 24 November 2007 (UTC)

Limit "in the space"
Under "Comments" it says: "The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence". Does the phrase "has a limit" always mean "has a limit in the space itself"? Sometimes this qualification is spelt out, as for example later in the same section of the present article, but other times it seems to be implied, for example under Complete metric space where it says that a certain sequence converges when "considered as a sequence of real numbers", instead of saying more simply that the sequence of rationals converges to an irrational. Under Cauchy sequence, which is the article referenced by the present one, the equivalent example is phrased in the simpler form, but a different example again says "If one considers this as a sequence of real numbers, however, ..." —Preceding unsigned comment added by G Colyer (talk • contribs) 17:51, 11 June 2008 (UTC)

examples
is it obvious that n^(1/n) = 1? —Preceding unsigned comment added by Slakov (talk • contribs) 17:16, 28 October 2008 (UTC)


 * If you take the log it is clear the ratio tends to zero. Katzmik (talk) 17:22, 28 October 2008 (UTC)

Limit of a number sequence
I think the article mast be renemed to «Limit of a number sequence». But I could not see the rename button. :( --OZH (talk) 07:33, 10 December 2010 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: Not moved. Jafeluv (talk) 14:27, 17 December 2010 (UTC)

Limit of a sequence → Limit of a number sequence — This article is concerned with a number. The aricle «Limit of a sequence» may propose a general aproach to some different types of limits for merical, topological and functional spaces. So, the article «Sequence» may be split onto two articles: «Sequence» (general) and «Number sequence». OZH (talk) 07:54, 10 December 2010 (UTC)
 * Oppose. This seems to be more of a split request than a rename request, and it doesn't seem appropriate to me.  As you can see, it includes limit of a sequence in topological spaces.  — Arthur Rubin  (talk) 09:14, 11 December 2010 (UTC)
 * Comment At the moment, the article talks about limits of sequences in any topological space. I've made a small change to the lead to make this more clear. Paul August &#9742; 14:39, 11 December 2010 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Requested cleanup/clarification
Hello. I am not adequately comfortable with the notation or the mathematics, so I do not want to perform this cleanup myself. I am therefore requesting that someone better qualified do it.

In the first section, "Formal Definition," under the very first bullet point, the symbol " xn " seems to function first as the name of a sequence (x1, x2 ...) and later as the name of a generic element of that sequence. Here is the confusing language:

A real number L is said to be the limit of the sequence xn, written


 * $$\lim_{n \to \infty} x_n=L, $$


 * if and only if for every real number &epsilon; &gt; 0, there exists a natural number N such that for every n &gt; N we have | xn&minus;L | &lt; &epsilon;.

Note that the definition of a limit given mentions subtracting a number L from "xn". This latter term refers to the nth member of the sequence rather than the sequence itself. Correct? Also the nested uses of the subscript "n" are rather confusing. — Preceding unsigned comment added by 98.232.15.46 (talk) 22:09, 30 June 2011 (UTC)


 * Correct, x_n in the definition is the nth term, not the sequence. I'm going to edit the article and use the sequence notation used in the article about sequences to differentiate the sequence itself from the nth term. Sunny642 (talk) 13:23, 9 August 2023 (UTC)

Convergence, Divergence and No limit
I thought a convergent sequence is a sequence with a finite limit, a divergent sequence is a sequence with an infinite limit, and all the rest are said to have no limit, in which case this article contains a few mistakes. (Oscillation (mathematics) is not behaviour of a divergent sequences for example. Brad7777 (talk) 17:54, 6 November 2011 (UTC)


 * That does not match the usage of the terms "convergent seqence" and "divergent sequence" that I am aware of. All sources I know define "divergent sequence" to mean "a sequence that does not converge". This includes bounded sequences with convergent subsequences that converge to different limits.
 * And to define the notion of an "infinite limit" would probably require some extra care, even if we only talk about sequences of real numbers. What about a sequence like 1, -2, 3, -4, 5, -6, ...? Does this sequence have one or two "infinite limits"? See also extended real number line and real projective line. — Tobias Bergemann (talk) 11:04, 7 November 2011 (UTC)
 * sorry the link should have been Oscillation (mathematics). A bounded sequence with subsequences that converge to different limits, is a way to tell a sequence has no limit, (for example the sequence -1,1,-1,1,-1,1.. has no limit)


 * the sequence 1,-2,3,-4,5,-6,... is said to have no limit
 * the sequence 1,2,3,4,5,6,... is said to diverge (to infinity)
 * the sequence -1,-2,-3,-4,-5,-6,... is said to diverge (to minus infinity)
 * A sequence that is a map from the natural numbers to the reals, can only have upto one limit, but a sequence from the integers to the reals may have two, (...-3,-2,-1,0,-1,2,3,...) or so i believed...
 * I think however all Series (mathematics) either converge or diverge, but sequences can have no limits Brad7777 (talk) 16:42, 8 November 2011 (UTC)
 * Sorry, that's one interpretation, but
 * the sequence 1,-2,3,-4,5,-6 converges to infinity on the projective line, but diverges (has no limit, in your notation) when considered on the extended real line.
 * And I've never heard of a potential difference between a series diverging and the sequence of partial sums diverging.
 * — Arthur Rubin (talk) 19:49, 8 November 2011 (UTC)

Brad7777 (talk) 22:10, 8 November 2011 (UTC)
 * A series diverging and a sequence of partial sums diverging is the same. My maths text book say thats a sequence can converge, diverge or have no limit (the same for a series too) thats all (Guide to Analysis (Mathematical Guides) by Mary Hart) Brad7777 (talk) 22:10, 8 November 2011 (UTC)

Weierstrass in the 1770s
The last line of the History section states that:

The modern definition of a limit (for any ε there exists an index N so that ...) was given by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, little noticed at the time) and by Weierstrass in the 1770s.

Weierstrass was not alive in 1770s (according to Karl Weierstrass); probably a typo (should have been 1870s?) — Preceding unsigned comment added by 212.235.33.45 (talk) 14:16, 11 January 2012 (UTC)

Restructuring
I heavily modified the structure of this article, splitting it into sections for sequences in the reals, metric spaces and topological spaces. It is not highly polished, but might be a better skeleton to hang things off. I didn't copy across everything from before that could reasonably be included, so there could be things in the previous version that people want to salvage. xnn (talk) 22:30, 8 February 2012 (UTC)

Miscaptioned graphics are not helpful
In the graphic that appears to be a rescaled and discretized version of a graph of y = x2 sin(x), the first sentence of its caption reads

"The plot of a Cauchy sequence (xn), shown in blue, as xn versus n."

This statement is self-contradictory. A Cauchy sequence is a different thing from its graph as a function of its index. So, there is no such thing as the plot of a sequence as the sequence versus its index.

The purpose of this section of the article should be to educate people who don't know what a Cauchy sequence is, or at least to remind them without their having to read the entire Cauchy sequence article. So a statement like this one in that section:

"A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large."

is extremely unhelpful, since it also makes no sense: What does "terms become arbitrarily close together" mean in the context of "n gets very large" ??? The index n is one index. The word "terms" is plural. So how is someone supposed to interpret the phrase "as n gets very large".

If the only references to a subject (in a larger article) are misleading, it's far better to just omit that section. But far better yet is to include a correct and clear explanation of the subject, even if it does not go into great detail.Daqu (talk) 01:38, 21 September 2014 (UTC)

"Failed to parse..." error
I did not change anything in the article. Just adding this note: The page was rendering with a big slab of red error-message-text in the middle of the "Examples" section: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle x_n = \frac1{n^2}} This error text did not appear when rendering the same wiki-markup source in "Preview" mode, so I "purged" the page according to https://www.mediawiki.org/wiki/Manual:Purge. ("Purging forces MediaWiki to clear the cached version of a given page, forcing the page to be redisplayed from its source"). Purging cleared the error text. --Dennis J au (talk) 01:44, 18 March 2017 (UTC)

More Common Limes Definition
I made an edit in section "Formal definition". Although N in |R and n > N also works, its more common to use N in |N and n>=N and it is less circular what the construction of reals concerns. This is also more consistent with the section "Infinite limits" and what is also found in the German wiki. See also here for this use: http://spot.colorado.edu/~baggett/chap2.pdf Jan Burse (talk) 08:12, 4 May 2017 (UTC)

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"Limit of a sequence"
The article, in my opinion, should explain right at the start that "limit of a sequence" is very different from "limit of a series" (see Series (mathematics) and Sequence). The limit of a sequence is the value that successive terms of a list of numbers created by an algorithm* converges to, if it does indeed converge to a single value (it may not, in which case the limit of the sequence does not exist).

The "limit of a series" is very different. It is the sum of the first n numbers in a sequence, considered as n approaches infinity from below or minus infinity from above.

For example, in the sequence 1,1,1,1,1,..., the limit of this sequence is 1, a constant. But in the corresponding series (infinite limit), the partial sums are 1,2,3,4,5,..., which is actually a definition of infinity, and therefore the series diverges. The limit of the partial sums does not exist except as the abstraction "infinity" (Cantor's countable infinity).

This shows clearly that even if a sequence converges to a value, the corresponding infinite series (sum of the numbers added so far) may easily diverge. And this shows that the two terms "limit of a sequence" and "limit of a series" are very different. They should be explained first, so newcomers to sequences and series can understand their different infinite limits free of confusion.


 * Technical Note: an algorithm must complete in a finite number of steps, by definition (see Algorithm). When, in our imagination, we compute an infinite sequence, it is possible that the algorithm may take more steps with each term in the sequence. In such a case, the algorithm will take increasing numbers of steps, and increasing time to compute, as the sequence approaches infinity. But since infinity cannot (by definition) be reached, the algorithm is guaranteed to complete in a finite number of steps for each term. Hence, "algorithm" is the correct description of how each number in the sequence is created (even when such creation is impractical).

David Spector (talk) 14:15, 11 November 2019 (UTC)

Introduction to Real Numbers section and use of notation
The introduction to the Real Numbers section states that a number is the limit of a sequence if the numbers of the sequence get closer to it and not any other number. I believe this is incorrect; consider the sequence {3/2, 5/4, 9/8, 17/16...}. While the limit of this sequence is 1, the numbers are also getting closer to, and will keep getting closer to, say, 0.99999 (or 0.5, or -17).

I propose the paragraph be rewritten like this:

"A number L is the limit of a sequence if one can provide a term x_n in the sequence such that their difference is within a specified neighbourhood of L, for all neighbourhoods of L."

I also propose the article use the notation used in article about sequences to denote a sequence instead of x_n to differentiate the sequence itself from the nth term, as the article currently uses x_n to denote the sequence itself sometimes and x_n to denote the nth term other times, such as in the mathematical definition. The aforementioned article uses (x_n)_n∈N to denote the sequence (x_1, x_2, ..., x_n), which I think is better than denoting the sequence (x_n) since it is used to denote the nth term at times in this article.

Please let me know what you think. Sunny642 (talk) 22:26, 8 August 2023 (UTC)