Talk:Limit of a sequence

message: 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) < epsilon for any epsilon > 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=−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)
thread_title: Really like this page

message: 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)
thread_title: Exercises.

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title: How?

message: 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)
thread_title: Basic Examples to help calculate limits more readily?

message: 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)
replies: text: 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)
replies: text:  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)
thread_title:  Wikipedia sucks for math 

message: 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)
replies: text: 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)
thread_title:  Encyclopedia should be accessible to laymen, this article is not. 

message: 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)
replies: text: 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)
thread_title:  Infinitely Small 

message: 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)
thread_title: Sequences with finite elements

message: 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)
thread_title:  p > 0, or p > 1? 

message: 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)
thread_title:  Informal language? 

message: 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)
thread_title:  Limit "in the space" 

message: is it obvious that n^(1/n) = 1?  —Preceding unsigned comment added by Slakov (talk • contribs) 17:16, 28 October 2008 (UTC)
replies: text: If you take the log it is clear the ratio tends to zero.  Katzmik (talk) 17:22, 28 October 2008 (UTC)
thread_title:  examples 

message: 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)
thread_title:  Limit of a number sequence 

message: 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 ε > 0, there exists a natural number N such that for every  n > N we have | xn−L | < ε.
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)
replies: text: 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)
thread_title:  Requested cleanup/clarification 

message: 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)
thread_title:  Weierstrass in the 1770s 

message: 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)
thread_title:  Restructuring 

message: 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)
thread_title:  Miscaptioned graphics are '''not''' helpful 

message: 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)
thread_title:  "Failed to parse..." error 

message: 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)
thread_title:  More Common Limes Definition 

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thread_title:  External links modified 

message: 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)
thread_title:  "Limit of a sequence" 

message: 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)
thread_title: Introduction to Real Numbers section and use of notation