1830:). 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).
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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
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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,
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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
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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,
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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
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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
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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.
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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, Knowledge can be a both a learning resource and a reference resource, but unfortunately, when it comes to
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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? Knowledge is a source
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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 = Lim (1-1/n) / Lim(1)
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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
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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),
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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
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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
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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
Knowledge, which is for reference as opposed to learning.
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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".
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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
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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 💕 that should help especially those who were not blessed with, or could not afford to build, a strong math background.
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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.
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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.
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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
311:= / = (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/n)/(1/n). If you do this you'll see that it converges to 4/9.--
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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
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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:
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using merely logic can throw u off. for instance (2n+1)/(3n-1) 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
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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
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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}}
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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"
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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
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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?
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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"
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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.
1795:). 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
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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 :
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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.
902:, 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, ..."
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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.
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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.
1043:— 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 «
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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/n p: -->
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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/n)=0 for p:
1653:=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:
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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...
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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:
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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.
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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
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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.
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I would try to edit it myself, but I'm still pretty new here at the
Knowledge and wouldn't be able to make it look nice. At least they can be here for anyone that may need them for now.--
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1859:"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."
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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)
1069:. 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. —
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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
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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.
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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:
1384:. 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)
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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.
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Knowledge 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 -
894:"? 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
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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.
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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.
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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.
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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). --
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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
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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
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In the graphic that appears to be a rescaled and discretized version of a graph of y = x sin(x), the first sentence of its caption reads
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Topological space? Limits of sequences are taught to beginning (FIRST YEAR) calculus students! What are you doing including this here? -
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This statement is self-contradictory. A Cauchy sequence is a different thing from its graph as a function of its index. So, there is
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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This error text did not appear when rendering the same wiki-markup source in "Preview" mode, so I "purged" the page according to
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to know what a limit is. And if they do for some reason like memory loss, then a first-year calculus definition is sufficient. -
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Some exercises, resolution of exercises and stuff like that could be interesting... and helpful for college homeworks :)
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https://web.archive.org/web/20040905075957/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html
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Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.
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Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.
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0, then p is the limit of {s sub n}. Really unclear. Thanks wikipedia. Sorry if I put this comment in the wrong place.
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And I've never heard of a potential difference between a series diverging and the sequence of partial sums diverging.
288:(yes, i am sort of asking for help of my hw, but i also think the article should be expanded to include that stuff.)
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so the limit is 1. but is there now "way" to do it as with the function? is it merely thinking about it logically?--
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Does “limit of a sequence” have any meaning when the sequence in question has only a finite number of elements?
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I think the article mast be renemed to «Limit of a number sequence». But I could not see the rename button. :( --
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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th member of the sequence rather than the sequence itself. Correct? Also the nested uses of the subscript "
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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The intro really helped me understand the definition since in my book it states that for some n: -->
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A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large.
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on
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also getting closer to, and will keep getting closer to, say, 0.99999 (or 0.5, or -17).
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which is the " largest possible number less than 1", since for any real number
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Prove if {s sub n} converges then {|s sub n|} converges. Is the converse true?
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so there could be things in the previous version that people want to salvage.
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http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html
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When you "do" a function, you DON'T plug in the value. The definition of the
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of knowledge. To remove some of that knowledge because
Knowledge is not the
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I made an edit in section "Formal definition". Although N in |R and n : -->
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is odd, then the absolute value has a limit but not the original sequence.
174:=N dependent on epsilon if d(p sub n, p) < epsilon for any epsilon : -->
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the sequence -1,-2,-3,-4,-5,-6,... is said to diverge (to minus infinity)
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Note that the definition of a limit given mentions subtracting a number
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Ah. sorry. I missed the "real" part and thought you just said number.
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except as the abstraction "infinity" (Cantor's countable infinity).
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Encyclopedia should be accessible to laymen, this article is not.
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Encyclopedia should be accessible to laymen, this article is not.
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either converge or diverge, but sequences can have no limits
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the sequence 1,2,3,4,5,6,... is said to diverge (to infinity)
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as the plot of a sequence as the sequence versus its index.
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A series diverging and a sequence of partial sums diverging
1047:» may be split onto two articles: «Sequence» (general) and «
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So the converse isn't true, but what about the proof part?
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for additional information. I made the following changes:
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If you take the log it is clear the ratio tends to zero.
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It would be nice to add also the notion of cesaro limit.
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Introduction to Real Numbers section and use of notation
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the sequence 1,-2,3,-4,5,-6,... is said to have no limit
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to differentiate the sequence itself from the nth term.
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The following discussion is an archived discussion of a
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N also works, its more common to use N in |N and n: -->
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is not behaviour of a divergent sequences for example.
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limited by five. Anyone wish to comment on the subject?
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The above discussion is preserved as an archive of a
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Basic Examples to help calculate limits more readily?
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I also propose the article use the notation used in
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The limit as n goes to infinity of a=0 if |a|<1.
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The limit as n goes to infinity of 1/n=0 for p: -->
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256:http://en.wikibooks.org/Calculus/Limits/Exercises
1856:I propose the paragraph be rewritten like this:
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1915:Knowledge level-4 vital articles in Mathematics
1710:This message was posted before February 2018.
1506:); probably a typo (should have been 1870s?)
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1220:{\displaystyle \lim _{n\to \infty }x_{n}=L,}
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1655:http://spot.colorado.edu/~baggett/chap2.pdf
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633:{\displaystyle 9.9999999999...=10N}
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276:ok, i'm still a little confused.
178:Here is an exercise from my book:
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976:Limit of a number sequence
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441:18:43, 28 March 2006 (UTC)
430:14:23, 28 March 2006 (UTC)
1382:Oscillation (mathematics)
1360:extended real number line
1336:Oscillation (mathematics)
1303:22:09, 30 June 2011 (UTC)
1287:" are rather confusing.
922:17:51, 11 June 2008 (UTC)
830:02:21, 27 July 2007 (UTC)
560:02:55, 8 April 2006 (UTC)
529:18:00, 5 April 2013 (UTC)
514:07:34, 19 June 2009 (UTC)
328:20:59, 6 April 2006 (UTC)
267:17:53, 5 April 2013 (UTC)
145:
78:
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1491:Weierstrass in the 1770s
1106:Please do not modify it.
1009:Please do not modify it.
816:11:11, 26 May 2006 (UTC)
807:07:36, 26 May 2006 (UTC)
790:19:37, 25 May 2006 (UTC)
776:23:18, 22 May 2006 (UTC)
743:22:52, 22 May 2006 (UTC)
582:Consider the following:
499:14:13, 22 May 2008 (UTC)
478:22:36, 25 May 2006 (UTC)
455:00:26, 23 May 2006 (UTC)
421:Knowledge sucks for math
404:22:43, 25 May 2006 (UTC)
152:project's priority scale
1865:article about sequences
1673:External links modified
1668:08:12, 4 May 2017 (UTC)
1311:article about sequences
109:WikiProject Mathematics
1895:C-Class vital articles
1626:
1565:), shown in blue, as x
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1783:"Limit of a sequence"
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1241:such that for every
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896:Complete metric space
796:non-standard analysis
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169:Really like this page
36:level-4 vital article
1789:Series (mathematics)
1722:regular verification
1403:Series (mathematics)
1401:I think however all
1364:real projective line
1179:
881:Limit "in the space"
764:)/2 is greater than
709:
675:
663:{\displaystyle 9=9N}
645:
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132:mathematics articles
1712:After February 2018
1682:Limit of a sequence
1037:Limit of a sequence
892:in the space itself
724:{\displaystyle N=1}
336:limit of a function
1816:are very different
1766:InternetArchiveBot
1717:InternetArchiveBot
1434:extended real line
1234:0, there exists a
1217:
1197:
890:mean "has a limit
861:Informal language?
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101:Mathematics portal
45:content assessment
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1512:comment added by
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900:Cauchy sequence
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1153:A real number
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1101:requested move
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73:High‑priority
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1736:source check
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1558:
1555:
1549:
1531:
1508:— Preceding
1501:
1497:
1494:
1476:
1459:
1446:Arthur Rubin
1425:
1332:
1295:98.232.15.46
1289:— Preceding
1284:
1280:
1275:
1271:
1267:
1265:
1259:
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1250:
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1242:
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1071:Arthur Rubin
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1035:
1018:
1016:
1008:
1001:
979:
931:
891:
887:
884:
864:
840:
824:
810:
793:
783:Infintesimal
780:
765:
761:
757:
753:
748:There is no
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671:
641:
611:
584:
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540:
537:
517:
503:
491:74.66.240.51
481:
469:
464:
461:
424:
416:
413:
410:Cesaro Limit
397:
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368:
301:
290:
277:
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253:
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224:
208:
203:
199:
198:is even and
195:
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107:
51:WikiProjects
34:
1634:Dennis J au
1086:Paul August
935:—Preceding
904:—Preceding
770:Paul August
750:real number
521:Unnachamois
485:—Preceding
295:Jaysscholar
293:confused.--
283:Jaysscholar
259:Unnachamois
227:—Preceding
123:Mathematics
114:mathematics
70:Mathematics
1889:Categories
1773:Report bug
1170:, written
846:metaprimer
800:hyperreals
341:Kprateek88
247:Exercises.
1828:Algorithm
1756:this tool
1749:this tool
1660:Jan Burse
1583:should be
1569:versus n.
1262:| < ε.
1249:we have |
1159:the limit
1019:Not moved
855:_1?": -->
827:Morris K.
813:He Who Is
787:He Who Is
740:He Who Is
562:Piepants
475:He Who Is
401:He Who Is
213:Thenub314
39:is rated
1873:Sunny642
1793:Sequence
1762:Cheers.—
1510:unsigned
1478:Brad7777
1464:Brad7777
1407:Brad7777
1340:Brad7777
1315:Sunny642
1291:unsigned
1045:Sequence
949:contribs
937:unsigned
928:examples
918:contribs
910:G Colyer
906:unsigned
868:Lex aver
557:Piepants
487:unsigned
353:Contribs
229:unsigned
1686:my edit
1600:detail.
1552:helpful
1354:limits.
1082:Comment
1023:Jafeluv
962:Katzmik
835:p : -->
506:Rckrone
207:=−1 if
150:on the
41:C-class
1449:(talk)
1270:from "
1074:(talk)
1067:Oppose
941:Slakov
888:always
569:Elroch
438:Elroch
194:=1 if
47:scale.
1820:first
1245:: -->
844:1. —
760:+ (1-
391:EK711
313:EK711
28:This
1877:talk
1839:talk
1791:and
1664:talk
1638:talk
1606:talk
1602:Daqu
1539:talk
1518:talk
1482:talk
1468:talk
1411:talk
1372:talk
1366:. —
1362:and
1344:talk
1319:talk
1299:talk
1057:talk
1027:talk
986:talk
966:talk
945:talk
914:talk
872:talk
850:talk
798:and
736:also
525:talk
510:talk
495:talk
470:only
452:Iopq
427:Iopq
347:Talk
272:How?
263:talk
237:talk
217:talk
142:High
1799:).
1730:RfC
1700:to
1550:not
1535:xnn
1426:con
1184:lim
1103:.
1053:OZH
1051:».
982:OZH
842:-->
802:. —
307:-->
278:How
1891::
1879:)
1841:)
1743:.
1738:}}
1734:{{
1666:)
1640:)
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1541:)
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1484:)
1470:)
1460:is
1444:—
1413:)
1374:)
1346:)
1321:)
1301:)
1260:L
1194:∞
1191:→
1141:,
1059:)
1039:→
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1021:.
1006:.
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947:•
920:)
916:•
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837:1?
785:.
625:10
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465:is
385:0.
375:0.
350:|
265:)
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180:1)
1875:(
1837:(
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1567:n
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1272:x
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1255:n
1251:x
1247:N
1243:n
1239:N
1215:,
1212:L
1209:=
1204:n
1200:x
1188:n
1167:n
1163:x
1155:L
1147:2
1143:x
1138:1
1134:x
1129:n
1125:x
1089:☎
1055:(
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984:(
964:(
943:(
912:(
870:(
848:(
773:☎
766:x
762:x
758:x
754:x
719:1
716:=
713:N
690:9
687:9
682:=
679:N
658:N
655:9
652:=
649:9
628:N
622:=
598:N
595:=
523:(
508:(
493:(
356:)
344:(
261:(
235:(
215:(
209:n
204:n
200:s
196:n
191:n
187:s
154:.
53::
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