Talk:Limit of a sequence

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[edit] Exercises.

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


[edit] 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)

[edit] 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.)

[edit] Cesaro Limit

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


[edit] 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)

[edit] 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)

[edit] 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 ☎ 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. —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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)