Talk:Geometric series/Archive 1

Archive 1

Some links for references.

After being momentarily confused by one of the images I thought I might improve it. Here are some links to "geometric" methods for finding the sum of a geometric series.

• Proof without Words: Geometric Sums by Warren Page. http://www.jstor.org/stable/2689632

• J. H. Webb, $\sum \limits _{n=0}^{\infty }ar^{n}={\frac {a}{1-r}}$, Proof without words, http://www.jstor.org/stable/2689568

• Proof without Words by Benjamin G. Klein and Irl C. Bivens. http://www.jstor.org/stable/2689356

• Proof without Words: Geometric Series by Elizabeth M. Markham http://www.jstor.org/stable/2690738 (I like one of the presentations here better then ours)

• Proof without Words: Geometric Series by Sunday A. Ajose and Roger B. Nelsen http://www.jstor.org/stable/2690617 (This contains almost exactly one of our images, should probably reference this.)

• Proof without Words: An Alternating Series by James O. Chilaka http://www.jstor.org/stable/2691280 (This one is very interesting because it is the only to address an r<0 that I have found so far.)

• Proof without Words: Geometric Series by The Viewpoints 2000 Group http://www.jstor.org/stable/2691106

That's all for now. — Preceding unsigned comment added by Thenub314 (talk • contribs) 14:19, 21 February 2009 (UTC)

proof:

I think the proof should be changed to reflect that s=1+r+..r^(n-1) versus rs=r+r^2+r^3...+r^n hence s = (r^n-1)/(r-1) —Preceding unsigned comment added by 192.18.192.76 (talk) 10:26, 6 May 2009 (UTC)

As the first line of this article says, it is about infinite sums. -- Jitse Niesen (talk) 10:47, 6 May 2009 (UTC)

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Does anybody have some information about the double geometric series, given by $\sum \sum _{m\geq 1,n\geq 1}x^{mn}=\sum _{m\geq 1}{\frac {x^{m}}{1-x^{m}}}$, with |x|<1? In particular, what happens if x is close to 1 Are there asymptotic formula available in that case? 140.112.50.253 (talk) 03:06, 8 May 2009 (UTC)

I believe that when x goes to 1, with 0 < x < 1 one has that the sum is equivalent to

${\frac {1}{1-x}}\ln {\Bigl (}{\frac {1}{1-x}}{\Bigr )},$

by comparing to Riemann sums for ${\frac {1}{1-t}}$ on the interval $[0,x],$ with division points $x,x^{2},x^{3},\ldots$ --Bdmy (talk) 13:10, 8 May 2009 (UTC)

To be more precise, I think that as x tends to 1, one has

${\Bigl (}\sum _{m\geq 1}{\frac {x^{m}}{1-x^{m}}}{\Bigr )}-{\frac {1}{1-x}}\ln {\Bigl (}{\frac {1}{1-x}}{\Bigr )}\sim {\frac {c}{1-x}}$

for some positive constant $c\sim 0.57\dots$ that I am not able to express in closed form. --Bdmy (talk) 13:30, 9 May 2009 (UTC)

Actually I am now sure that $c$ is equal to the Euler-Mascheroni constant, and that the expansion above must be written somewhere, and extended to more terms. --Bdmy (talk) 14:47, 9 May 2009 (UTC)

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on Zeno's paradox:

The hidden assumption is that a sum of infinite number of finite steps can not be finite. This is of course not true as evident by the convergence of the geometrical series with r=1/2 illustrated at the picture at the introduction section of this article.

I don't think this is right. That a finite distance can be divided into infinitely many steps is not a hidden assumption, it is the main premise of the argument. The paradox (to show why this is absurd) is that it would take an infinite amount of time to complete these steps - here is where the real "hidden" assumption comes in - that each step requires a finite and bounded amount of time and therefore the whole process would take forever. Zeno's paradox is resolved if it is assumed that the time to perform each step can be made infinitesimally small, allowing convergence.

N. —Preceding unsigned comment added by 70.49.87.164 (talk) 04:45, 1 July 2009 (UTC)

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Formula in section 2.2

The formula in section 2.2 says it is "the sum of the first n terms of a geometric series". It actually is the sum of the first n+1 terms of a geometric series.

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—Preceding unsigned comment added by 88.81.99.141 (talk) 17:17, 9 April 2010 (UTC)

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Introductory Series Expression and Series Illustration Do Not Agree

Please consider either modifying the 1/2^n series expression or the 1/4^n series illustration so both represent the same series or specify the 1/4^n series in the illustration's caption. Persons using this page to acquaint themselves with the subject will naturally associate the expression and the illustration. The fact that 1/2 appears as the first term in the expression and as a dimension in the illustration could be especially confounding. Thank You. VoilàY'all (talk) 23:52, 25 June 2010 (UTC)

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NOTE: I subsequently added information to the illustration caption to help distinguish series expression in the text from the series summation in the illustration. VoilàY'all (talk) 00:26, 26 June 2010 (UTC)

Archimedes' quadrature of the parabola proof

Actually there is error in image. If triangle area is 1, then scale is not 1 under triangle, but another. With Heron's formula it apears that if a=b=c=1, then area $S={\sqrt {{1+1+1 \over 2}\cdot (1.5-1)\cdot (1.5-1)\cdot (1.5-1)}}={\sqrt {1.5\cdot 0.5^{3}}}={\sqrt {0.1875}}=0.433012701.$

The area enclosed by a parabola and a line is the union of infinitely many triangles.

If area is 1, then there is quick way to find a=b=c values by looking how much times is different area 1 from area 0.433, so difference is 1/0.433012701=2.309401077. So normaly if triangle have a=1 and h=1, then his area S=0.5*1*1=0.5, and if triangle have a=4 and h=4, then triangle are is 16 times bigger, S=0.5*4*4=8. Or if triangle a=2 and h=2, then S=0.5*2*2=2, so 4 times bigger. Thus if area is 2.3 times bigger, then $a={\sqrt {2.309401077}}=1.519671371$ times bigger, so this is exact a=b=c=1*1.519671371=1.51967 value. (for more approving this thinking let's say a=2, h=4, s=0.5*2*4=4 and if [lines are 2 times longer] A=4, H=8, S=0.5*4*8=16, value is 4 times bigger; or if a=4, h=2, s=0.5*4*2=4, by multiplying A=a*3=4*3=12, H=h*3=2*3=6, S=4*9=0.5*12*6=36, so 36/4=9 times, 3*3=9, get it?)

So to proof this series

$1\,+\,2\left({\frac {1}{8}}\right)\,+\,4\left({\frac {1}{8}}\right)^{2}\,+\,8\left({\frac {1}{8}}\right)^{3}\,+\,\cdots .$

need to find h, which with pythagor theorem is easy, because I know now, that a=b=c=1.519671371. So half of a is d=a/2=1.519671371/2=0.759835685. So by Phytagor theorem $h={\sqrt {b^{2}-d^{2}}}={\sqrt {1.51967^{2}-0.7598^{2}}}={\sqrt {1.732050807}}=1.316074013.$ So what kind of parabola in image is? It is $y=-x^{2}+h=-x^{2}+1.316074013$ parabola. Values x are [-0.759835685; 0.759835685] and values y are [0; 1.316]. By integrating I should get area S>1, so:

$S=\int _{-0.7598}^{0.7598}(-x^{2}+1.316074013)dx=(-{x^{3} \over 3}+1.316074013\cdot x)|_{-0.7598}^{0.7598}=$

$=(-{(-0.7598)^{3} \over 3}+1.316074013\cdot (-0.7598))-(-{0.7598^{3} \over 3}+1.316074013\cdot 0.7598)=$

$=(0.146230445-1)-(-0.146230445+1)=1.707539109$.

Also I want to try another integration way by finding area under parabola "lines" and then substracting it from area s=a*h=1.519671371*1.316074013=2. So parabola then is $y=x^{2}$. And so integration is simplier:

$S=\int _{-0.7598}^{0.7598}x^{2}dx={x^{3} \over 3}|_{-0.7598}^{0.7598}={(-0.7598)^{3} \over 3}-{(0.7598)^{3} \over 3}=-0.292460891.$

So P=s-|S|=2-0.292460891=1.707539109, this is the area under the parabola like in image.

Sides a, b and c may not be equal length (just looks so in wrong image). But why in article no information about What parabola it is? Maybe it's y=ax*x+c, maybe y=x*x+c, maybe y=x*x+bx+c, hm?

Anyway, Let's see what area under parabola in image will be with those series:

$P=1+2\cdot {1 \over 8}+4\cdot {1 \over 8^{2}}+8\cdot {1 \over 8^{3}}+16\cdot {1 \over 8^{4}}+32\cdot {1 \over 8^{5}}+64\cdot {1 \over 8^{6}}+128\cdot {1 \over 8^{7}}=$

$=1+0.25+0.0625+0.015625+0.00390625+0.000976562+0.00024414+0.000061035=1.333312988.$

So if I flip parabola in image and put it highest point onto point (x; y)=(0; 0), then I will have parabola y=x*x. So then I just will find y highest point with puting into x farthest point $y=0.759835685^{2}=0.577350268$ (because seems this series should be correct even if for triangle if a=b=c). So now h=0.577350268. And triangle area should be 1. So S=a*h=1.519671371*0.577350268=0.875823827. Somthing wrong, it apears that triangle is too small with area s=a*h/2=0.437911913.

I think found where is problem. In article there no any information what kind of parabola it is. If it is y=ax*x+bx+c then all calculations becomes wrong. Just why in article no information for what kind parabola you will get, you kinda make parabola around any triangle, calculate parabola area but don't know what kind this parabola is, this is problem.

Let's try to proof in another way. Let's take parabola, which I choosing, say I TAKING PARABOLA y=x*x and x==[-1; 1] and y==[0; 1]. So triangle inside parabola have base a=2 and h=1. Triangle area is s=a*h/2=2*1/2=1. Now with integration method I will find area under parabola lines:

$S=\int _{-1}^{1}x^{2}dx={x^{3} \over 3}|_{-1}^{1}={-1 \over 3}-{1 \over 3}=-{2 \over 3}=-0.66666$. So Rectangle area is S=a*h=2*1=2. And so parabola area is [rectangle area minus area under parabola lines] P=a*h-|-0.6666|=2-0.6666=1.333333.

Now by using series I will try to get same answer and it seems that I previously got it (1.333312988) and triangle area is equal to 1. So those series are now proved. And it means, that those series only good for parabola y=x*x.

Now I just want to see, is it correct for parabola y=x*x, when x==[-3; 3] and y==[0; 9]. But then first need to find triangle area of this kinda big parabola. So a=6 and h=9, so triangle area is s=a*h/2=6*9/2=27, so triangle area is 27 times bigger than in previous case. So Parabola area should be P=1.33333*27=35.99999=36. So let's check it by integrating:

$U=\int _{-3}^{3}x^{2}dx={x^{3} \over 3}|_{-3}^{3}={(-3)^{3} \over 3}-{3^{3} \over 3}=-9-9=-18.$

So here now I got area |U|=18 under parabola lines. Rectangle area is twice bigger than triangle R=a*h=6*9=54. And so parabola area from point (x; y)=(0; 0) to highest point [also [to] triangle base] is P=R-|U|=a*h-|U|=54-18=36. So just perfect, answer the same. Geometrical series for parabola area finding completely proved. — Preceding unsigned comment added by 84.240.9.58 (talk) 14:00, 11 August 2010 (UTC)

Geometric View

I removed the section below. It is unclear whether the author is addressing a mistake in the figure as it appears here or as it appears in the cited text. Moreover, how the figure relates to the geometric sum is not self-explanatory, and so should be elaborated upon if re-included.

Here is a geometric way of looking at the geometric series from E.Hairer and G.Wanner, Analysis by Its History, section III.2, FIGURE 2.1, page 188, Springer 1996 - there is a mistake in the figure: it should read r/(1-r), then add the missing term 1 to get 1/(1-r) :

For completeness, when $r=1$, the sum of the first n terms is:

$a+a1^{1}+a1^{2}+a1^{3}+\cdots +a1^{n-1}=\sum _{k=0}^{n-1}a=an.$

Austinmohr (talk) 07:44, 25 May 2011 (UTC)

The "economics" section

By my reckoning the output at the bottom of the "economics" section should be 100*(1+I)/I, as opposed to 100/I as it currently stands. — Preceding unsigned comment added by 67.189.78.224 (talk) 00:54, 16 October 2011 (UTC)

Merger proposal

I propose that Geometric progression be merged in part or whole into Geometric series. The concept of the progression or sequence is necessary to understand the series, so it is necessary in the Series article. For some reason, the series article concerns infinite series, but finite series are described in the geometric progression article. This is confusing because it makes it sound as though Geometric series refers to the infinite progression, and never a sum of a finite sequence. Thelema418 (talk) 02:28, 24 August 2012 (UTC)

Althought a geometric progression is very different to a geometric series (one is a sequences, the other is a series), most of the geometric progression article seems to talk about geometric series. Provided Thelema418 is prepared to conduct the merger that s/he has proposed, I have no reservations about a merger taking place. — Fly by Night (talk) 22:19, 25 August 2012 (UTC)

Let us not to repeat the mistake with square (algebra). The proposed direction of a merger is egregiously incorrect. "Progression" is algebra, one can understand progression without infinity, and progressions are really used without any convergence, e.g. as polynomial rings. "Series" is a calculus and, in some sense, topology. These mathematical structures are very different. Of course, the current "geometric progression" has a lot of off-topic, but this should not encourage users to make articles progressively worse, as was made a year ago with redirecting square (algebra) to square number. Incnis Mrsi (talk) 09:13, 10 September 2012 (UTC)

I agree. Geometric progressions and geometric series belong to different parts of mathematics, and should be kept separate for roughly the same reason that the Addition and Series (mathematics) articles are separate. In particular, this article is part of Category:Calculus, while the Geometric progression article is not.Jim.belk (talk) 04:56, 26 January 2014 (UTC)

I would agree; they should be together as are Arithmetic Progression and Arithmetic Series. It's confusing to have Geometric Progression and Geometric Series as separate, lengthy articles. Startswithj (talk) 18:48, 30 November 2013 (UTC)

I don't think the analogy with Arithmetic Progression and Arithmetic Series is relevant. This article is primarily about infinite geometric series, which is the most common use of the term "geometric series" in mathematics. Indeed, most mathematicians would refer to the sum of a finite geometric progression as a "geometric sum" or a "finite geometric series". There is no concept of an "infinite arithmetic series", which is why there's no corresponding article on Arithmetic Series. Jim.belk (talk) 04:56, 26 January 2014 (UTC)

Really? No concept? (Also, finite geometric series are used all the time in computer science. But I agree that this article should primarily focus on the infinite case and that it's a step higher in mathematical sophistication from geometric sequences, justifying separate articles.) —David Eppstein (talk) 06:39, 20 December 2014 (UTC)

I also agree with the last few comments on this proposal, but for different reasons. Those people that are taking classes and that are new to this kind of math would likely be confused by the merger. High schools students taking math and looking this subject up on Wikipedia would not understand the difference. I know that I myself would be incredibly confused if these two completely different subjects were on the same page. Leon13552(talk) 13:12, 11 February 2014 (UTC)

Leon13552: They're not completely different. You can't define a geometric series without explicitly or implicitly using a geometric progression, though you can derive results about the later without references to the former; in that sense geometric progressions are more elemental than geometric series (but not that that is not a reason to split them any more than it is to have different articles for sum over each of N, Z, Q, R, C).

You mention understanding; however, in my view that's a nefarious generalization about secondary education students, and I think that even for that matter, those students who don't understand geometric series would still understand the part about geometric progressions in this article after the merger (they can just ignore the rest). Furthermore, given that the main use of geometric progressions in for defining geometric series, a unified article about both would make it easier to understand geometric series (for those who don't understand them already, of course) than 2 separate articles which mostly overlap, see also my comment below. Having an unified article may help to clarify the relation between both concepts. Thanks for contributing your viewpoint. Mario Castelán Castro (talk) 01:13, 8 February 2015 (UTC).

Agree with merge proposal: The current article geometric progression talks mainly about geometric series and geometric series already talks about geometric progressions. They are almost entirely duplicate in scope, and as a consequence they're also almost entirely duplicate in what their content covers, but with artificial content differences (specific way in which the ideas are presented, wording etcetera); this is bad for everyone: readers should read both articles to make sure they're not missing something important; the editors are faced with the situation of maintaining 2 explanations of the same topic with only arbitrary differences (like editing the articles covering the same topic in 2 separate encyclopedias). Both readers and editors waste effort and don't gain (or allow others to gain) any more insight about either geometric progressions and geometric series with this split. Also, the main use of geometric progressions (over complex numbers) is to sum them, I.e: to use them in geometric series.

For geometric progressions over other algebraic structures (finite fields, etcetera), we should have a separate article. There's little in common between geometric progressions over a finite field and over the complex numbers, both in what areas they are used (abstract algebra, cryptography for the former, analysis and a lot of real phenomenons exhibiting exponential growth or decay) and their properties (in the finite field, a geometric progression doesn't shrinks and the concept of a Cauchy sequence is undefined there). I noticed that the discussion is some years old. Do the people who oppose merging are still interested and hold that posture?. Regards. Mario Castelán Castro (talk) 00:45, 8 February 2015 (UTC).

The only way this merger proposal will make sense is if this page remains predominantly an article about Geometric Series with a minor side-note added way down the page on Geometric Progression (which would still eliminate the redundant work without making productive use of this article more difficult). That is because Geometric Series is one of the most dynamically useful concepts in calculus, with profound applications in everything from Industrial Engineering to Computer Science to Signal Processing to Taylor Expansions, whereas most users of this page are going to have approximately as much interest in Geometric Progressions as they are in a lecture on ear wax. Ahfretheim (talk) 04:23, 16 April 2015 (UTC)

The case r = 1

I reverted the recent addition of a finite sum when r = 1 that was put in by request since this was out of place in this article. The section was deriving the formula for the sum of a geometric series and started with the sum of the first n terms of such a series. This partial sum was a means to an end and not the subject of the section. The restriction that r ≠ 1 is rather important for the topic of this page and putting in this simple case does not add anything to the subject and can cause confusion. This should have been pointed out to the requester rather than just complying with the request. --Bill Cherowitzo (talk) 23:33, 28 October 2016 (UTC)

@Bill Cherowitzo : thanks. I missed the context, apparently. But why is there a long re-derivation of the partial sum formula here, then? It should be done elsewhere... So I look at Geometric progression, and see a section "Geometric series", that, currently, start with "A geometric series is the sum of the numbers in a geometric progression. For example:"... and then proceeds with the computation of partial sums! $n$ is not even defined in this section. There is some stuff on actual series later on. And (back in this article) why is there a subsection on "Generalized formula" in the same "Sum" section, that just gives a partial sum formula and does zilch with it? I think saying that the page has a "topic" and the section a "subject" are acts of sheer optimism. It's anything but clear (at least to me, and at a glance) what both these articles, or their sections, are trying to achieve. Merge or no merge, those pages need work. — Gamall Wednesday Ida (t · c) 00:12, 29 October 2016 (UTC)

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Wrong wording?

On the example at the start where the factor is 2/3 it says "Consider the sum of the following geometric series:". Is the wording wrong? Can you have a sum of a series? or should it say "Consider the sum of the following sequence". Or since they are putting s = 1 + 2/3+ ... should it just say "Consider the following series"? I am not advanced enough in math to be confident in changing the text.

-Adam Siembida

The "sum of the series" is the standard word used in this case. It requires a definition, that you will find in the article Series (mathematics). --Bdmy (talk) 09:24, 15 May 2009 (UTC)

No, it is not the wording that is wrong. It is the proof that is wrong. The example discussed is technically wrong. It assumes that the sum of the series exists in order to prove that the sum of the series exists. You cannot take (1 + 2/3 + 4/9 + ...) - (2/3 + 4/9 + ...) and assume that this is well defined. It could be Infinity - Infinity. This needs to be rewritten with partial sums, and take the limit of the partial sum. The n+1 term which remains in the difference of the nth and n+1th partial sum will approach 0 as n goes to infinity. — Preceding unsigned comment added by 69.92.54.48 (talk) 03:08, 19 November 2019 (UTC)

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Why the factor a everywhere?? --85.5.93.40 (talk) 22:28, 7 March 2011 (UTC)

This is how series are often presented in Calculus texts. While students should just be taught to factor out "garbage" like factors of a, I don't think a rewrite of the article is useful. Austinmohr (talk) 07:49, 25 May 2011 (UTC)

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Your article conflicts with geometric progression

Ooops ! I'm wrong this is really a serie.

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Maybe you mean "a series"? "Series", with a final "s", is either singular or plural in English; there is no word "serie". -- Mike Hardy

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Thanks I did'nt know. "Une série", "des séries" I believe it was the same in English. :)

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I copied over the piece on geometric series in geometric sequence. It still needs to be integrated into the article. On further thought I feel that Geometric sequence and Geometric series should be combined since they deal with each other. Therefore I'm putting them together under Geometric progression and redirecting.

-Mathematical error- The series is given as a sum from k = 0 to infinity of r^k for |r|<1 but when r is zero the first term would be 0^0 which is undefined. it should be 1 + a sum from k = 1 to infinity of r^k for |r|<1

-Please try to make this understandable to non experts The use of strange symbols with no definition is a drag for those of us who are not educated in math-language. Please give a definition of that greek symbol (is it sigma? or something else) - and not in mathese - but in regular language. Since this an encyclopedia and not a technical journal please keep the hermetic language to a minimum. Thankyou — Preceding unsigned comment added by Canuckistani (talk • contribs) 21:43, 5 January 2009 (UTC)

Zeno

The convergence of an infinite series does not nessesarly imply a solution to Zeno's paradoxes. The fact that the series converge does not explain how one may do an infinite ammount of steps in finite time. Just "doing" the mathematics does not solve the problem. i.e. just walking does not resolve the argument, even less a simple mathematical solution. i.e. one does not have, in actualitly, anything or anybody that can continue to count to infinity, or do an infinite task: one has restrictions in the real world that imply the mathematical idealization does not correspond to physical reality... 186.26.113.114 (talk) 18:35, 13 December 2019 (UTC)

Animal wall painting

A painting depicting an animal story. A mathematical geometric series of 0.5 - Xarunta Dhaqanka ee Hargeysa

Whole painting.

A wall painting in a cultural centre in Hargeisa, Somaliland, showed a geometric series. After some discussion, this photograph obtained OTRS-permission from Wikimedia. I would like to show this nice summation and was requested to mention this here. Thanks, Hansmuller (talk) 07:37, 10 January 2020 (UTC)

What information about geometric series do you expect that a reader of this article might learn from this image? In what way is the geometric series, specifically, central to the meaning of the image, rather than merely being an arbitrarily chosen mathematical formula chosen only for its decorative value? —David Eppstein (talk) 07:53, 10 January 2020 (UTC)

I agree with David Eppstein: What does the content of the painting have to do with the series? How does this image improve the article?—Anita5192 (talk) 16:13, 10 January 2020 (UTC)

I am the author of the artwork and the related article "The Zeno's Paradox in Somali Culture and other stories" (2015), and I am not sure what the content of the painting can add to this specific article, but my aim was to highlight the importance of the indigenous knowledge, and to promote the use of such knowledge in the formal school curricula in Somaliland. The aim of this particular paint is to introduce the role that ethnomathematics may have in the mathematical curriculum in the Horn of Africa. Instead of using the classical race between Achilles—a very fast runner, and a lowly tortoise, I proposed a well known somali fable in Somaliland which goes:

One day, the wild animals' family killed a she-camel for prey and the lion, the king of the family, assigned the hyena, which was considered the most idiotic member of the family, the task of distributing the prey to the beasts. The hyena divides the meat into two equal parts, one half for the lion and the other half for the rest of the family. The lion, displeased with this decision, punishes the hyena, injuring its eye by striking him with his front leg. The fox (considered the most intelligent and opportunist member of the family) is then assigned the task of dividing the prey amongst the family. The fox observes the situation and glances at the lion, which replies by striking his canine teeth together sharply; the fox says: "One half of the camel is for the king, the lion; from the remainder, again one half is for the king; again from the remainder one half is for the king; and so on." The lion, satisfied for this statement, asked the fox.

"Where did you learn this fairness and justice?"

"Here! When I saw the injured eye of the hyena," fox replied.

Somali children listening at this plane fable are hardly persuaded by the narrator that the lion got the whole camel. The narrator guesses that adding, [times by times], one half of the previous half to this one, without stopping the procedure, the result will be the unit, but he has no mathematical formula and demonstration procedure to explain that fact mathematically to the children. They cannot understand the underlying notion of infinity and the narrator is unable to give a theoretical frame to his intuitions.

This fable and others appeared on here: . I hope this helps the conversation here. Jama Musse Jama 05:38, 17 January 2020 (UTC) — Preceding unsigned comment added by Jmgurey (talk • contribs)

And by the way, I do not know who shot the photo of my paint and uploaded the image on wikipedia, and I am happy that it happened. I was not aware of the discussion for copyright permission at the moment it was proposed to be deleted for 'no permission' and I am happy it did not happen, and the image is on the public domain. Jama Musse Jama 05:44, 17 January 2020 (UTC) — Preceding unsigned comment added by Jmgurey (talk • contribs)

References

incorrect article header "This article is about infinite geometric series. For finite sums, see geometric progression."

I don't know how to change that first line of the article, but it appears to be incorrect. A geometric progression (or geometric sequence) is not a finite sum version of a geometric series. Gj7 (talk) 21:21, 20 December 2020 (UTC)

The template was incorrect, so I removed it. I also inserted a line into this article and the Geometric progression article explaining the distinction between the two, which are frequently confused.—Anita5192 (talk) 22:33, 20 December 2020 (UTC)

Perhaps one day the geometric progression article should be merged into this geometric series article, but I would not suggest that until this article reaches feature article status. That is far away but I think there is a path to get there. Gj7 (talk) 00:17, 21 December 2020 (UTC)

consistent name for a: common scale or first term?

I prefer "common scale" over "first term". In the article's formula section, the geometric proof of the geometric series closed form formula employs a partial geometric series S = ar^m + ar^m+1 + ... + ar^n-1 + arⁿ for any m < n. I think "first term" incorrectly implies there is something special about the case of m = 0. Instead, I prefer to think of multiplying or dividing a partial geometric series by r as "shifting" up or down the geometric growth curve defined by a and r even if m is much less than or much greater than zero. Gj7 (talk) 22:20, 18 December 2020 (UTC)

I just cited three sources for the section. None of these sources have a name for a, so I removed the phrase "first term" altogether.—Anita5192 (talk) 04:25, 19 December 2020 (UTC)

Cool. The article also has a couple "start term" names for a. Gj7 (talk) 05:09, 19 December 2020 (UTC)

I cleaned it up a little. "First term" is in several places in the article and would be tedious to remove, but "start term" was very sloppy, so I removed it.—Anita5192 (talk) 05:41, 19 December 2020 (UTC)

@Gj7:I recommend that you remove all references to the phrase, "common scale", unless you can cite it with a reliable source. As I wrote above, I have not seen any source naming the constant a, I don't think it is wise for us to invent nomenclature that is not in reliable journal papers or textbooks, and I don't see any reason to name it at all. I think the section "Common scale" that you just inserted should be renamed and simply refer to the constant as a.—Anita5192 (talk) 03:30, 22 December 2020 (UTC)

How about "common coefficient"? The geometric series is a special case of the power series and there are plenty of references to power series coefficients. What I am trying to avoid is a single letter variable that has different meanings in similar contexts. We can already see that with r which in the geometric series article is common ratio but r in the exponential growth article is growth rate. So r in one article is r + 1 in another, even though geometric growth and exponential growth are equivalent.Gj7 (talk) 04:53, 22 December 2020 (UTC)

Or better yet, how about "coefficient a"? Using a classifier avoids the controversy of naming and also avoids the ambiguity of a one letter reference. It seems like referring to "coefficient a" would be no more controversial than referring to "variable x". At the risk of bruising the ego of a :*), I don't think it is popular enough to pull off a one word celebrity name, let alone a one letter celebrity name. Kidding aside, according to Google Ngram, the phrase "exponential growth" is 43 times more popular than the phrase "geometric growth" even though they are mathematically equivalent.Gj7 (talk) 13:32, 22 December 2020 (UTC)

consistent notation when writing a partial geometric series: is r's highest power n-1 or n?

In the article's Formula section the highest power of r is n-1, but in the article's Proof of convergence section it is n.

In the geometric proof (in the caption of the newly added image), the highest power of r is n (not n-1) because it emphasizes that the closed form's power of r is one greater than the highest power of r in the series. However, if there is an editorial effort in this article to use consistent notation for the highest power of r and that consistent notation is decided to be n-1, I will edit the geometric proof to be consistent with that choice.Gj7 (talk) 15:44, 19 November 2020 (UTC)

In the closed-form formula derivation, I changed the highest power of r from n-1 to n, which matches the indexing used in the power series article.Gj7 (talk) 04:19, 31 December 2020 (UTC)

Is article's Sum/example subsection needed?

It seems to be a warm-up to the proof, but I think the proof is easy enough to not need a warm-up. Gj7 (talk) 19:20, 28 December 2020 (UTC)

Also, the Sum section's leading description of convergence might cause confusion: "The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series containing infinitely many terms." A counter example to this description of convergence is the harmonic series (i.e., 1 + 1/2 + 1/3 + 1/4 + ...) which also has terms that "become insignificantly small" but the harmonic series does not converge.Gj7 (talk) 19:24, 29 December 2020 (UTC)

After a 2-week wait period for some comment about the value of keeping the article's sum/example section, I've deleted it including the visual proof image. It was not clear to me how the image represented the terms of the series: was it the area of the circle, or the area of a sphere, or perhaps the volume of a sphere? Also, it seems as if the drawn circles decreased linearly instead of geometrically.Gj7 (talk) 14:16, 11 January 2021 (UTC)

Article good enough yet to be a good article?

I've nominated this article for a "Good Article" review. There is still work to do but I think the article has progressed beyond its current "C" rating and the feedback from a "Good Article" review might be very helpful in making it better.Gj7 (talk) 00:28, 15 January 2021 (UTC)

GA Review

This review is transcluded from Talk:Geometric series/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: David Eppstein (talk · contribs) 00:34, 17 January 2021 (UTC)

Although the nominator has been doing extensive work on improving this article, and it's on an important topic that I would like to see reach GA status, I think it is still far from there. The organization of sections is sloppy; for instance, why is the historical note about Euclid listed as an application (WP:GACR 1a)? The formatting of the mathematics is very inconsistent (some in html, others in LaTeX; GACR 1a). Although this is a technical subject, I am not at all convinced that efforts have been made to make it as understandable as possible to as wide an audience as possible (GACR #1a). Much of the material in the lead (such as the definition of a geometric series and the claim that these played important roles in the development of calculus) is not expanded later, and conversely most of the rest of the article is not appropriately summarized in the lead (GACR #1b). Most sections, and most paragraphs within the remaining sections, are unsourced, making it very far from GACR #2. Of the eight footnotes that are given (for an article with 14 subsections!), the first two are duplicates of each other, one is not a reliable source, one (the Abramowitz and Stegun citation) has a broken link, and the formatting is far from consistent (GACR #2). Some sections, such as the proof of convergence, appear to me to go into excessive detail for an encyclopedia article (GACR #3b). The 70-line caption to one of the figures is a serious problem with respect to GACR #6b, and the redundant images of cumulative sums also appear problematic with respect to this criterion. As such, I think this was unready for GA nomination, and meets criterion 1 of WP:GAFAIL. —David Eppstein (talk) 00:34, 17 January 2021 (UTC)

Rationale for not setting a = 1?

I see no reason at all to let the constant a be anything other than 1. The more "general" case a ≠ 1 is obtained by multiplying everything by a. I suggest removing all mention of a. Ulrigo (talk) 08:15, 16 April 2021 (UTC)

Yes, it is good to simplify by removing the coefficient a. It is also good to keep coefficient a to emphasize the similarity with the Taylor series and the Fourier series that have coefficients that cannot be removed. A compromise that I like is S/a = 1/(1-r). Gj7 (talk) 23:00, 24 April 2021 (UTC)

We follow reliable sources. Every source I've seen on this uses a constant, generally called a.--Salix alba (talk): 01:08, 25 April 2021 (UTC)

I just checked in four textbooks. In one textbook, the series has the common factor, a; in the other three, the series does not. However, I would prefer to leave the article as is, using the common factor, a, partly because the initial statement in the Coefficient a section is already cited, and partly because this form is more general: Without the common factor, a, some readers might think a geometric series must begin with 1.—Anita5192 (talk) 02:16, 25 April 2021 (UTC)

There is absolutely nothing more "general" about the case a ≠ 1. You just multiply the formula by a, and there you have your more "general" expression. By your logic, Pythagoras' theorem should be stated as ax² + ay² = az² for an arbitrary constant a. Young's inequality should be stated as axy ≤ ax²/2 + ay²/2 for every a > 0. And so on. Salix alba, are you insinuating that there are no reliable references for the case a = 1? Ulrigo (talk) 19:41, 27 April 2021 (UTC)

If I don't hear any further protests, I will rewrite the article to set a = 1, and comment on the "general" case a ≠ 1, in a couple of weeks. Ulrigo (talk) 16:28, 18 May 2021 (UTC)

The name geometric series indicates each term is the geometric mean of its two neighbors, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighbors. For consistency, are you planning to also rewrite the arithmetic progression article so that the first term is 1?Gj7 (talk) 19:51, 18 May 2021 (UTC)

Zeno's paradox

It seems to me that Zeno of Elea should be included in the article's Historic insights section. However, the current Zeno's paradoxes section under Applications is dismissive of the contributions by Zeno of Elea, making the claim that he made a mistake in assuming that the sum of an infinite number of finite steps cannot be finite. According to my old calculus book, "The ancient Greeks thought that no infinite set of numbers could possibly have a finite sum. Because of this feeling, they were caught in some logical paradoxes. Perhaps the most famous of these was given by Zeno of Elea (c. 500 B.C.). He pointed out that it is logically impossible to walk from one place to another. He reasoned that before a person could go the entire distance, d, he first has to walk half of d. Then, of the distance d/2 that remained, he had to go half of that, leaving a distance d/4 yet to be covered. But he would have to go half of that distance." And so on, showing that there are an infinite number of halves of distances even within a short walk, which contradicted the prevailing wisdom that the sum of an infinite number of finite steps could not be finite.

So instead of making the claimed wrong assumption, it appears that Zeno of Elea instead was giving a counter-example to a widely held wrong assumption. Also contradicting the dismissive tone, some well known scientists speak very highly of Zeno of Elea's contributions. This article has the quote, "Bertrand Russell once remarked that "Zeno's arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own." that references its source as a 1959 book "The History of the Calculus and its Conceptual Development" by Boyer.

In a couple weeks, if no one can support the current dismissive tone towards Zeno of Elea, I'll rewrite the section and move it to the Historic insights section. Gj7 (talk) 01:46, 13 May 2021 (UTC)

I agree that the section on infinitesimals should be rewritten or removed, but I don't see the issue with the first sentence. Whether Zeno intended his "paradox" as a demonstration of the falsity of the assumption that an infinite sum is infinite, or as an amusing riddle, or "that motion is nothing but an illusion" (quote from the main article on Zeno's paradoxes), is a question of interpretation. The way I read it, the first paragraph takes the same view as the main article. Ulrigo (talk) 16:26, 18 May 2021 (UTC)

I moved the description of the contribution by Zeno of Elea from the article's application section to its historic insights section. I intentionally left out any description of the regressive form of his dichotomy paradox because that tangent seems to be more suited for a philosophical riddle than to a description of the geometric series. Gj7 (talk) 17:22, 29 May 2021 (UTC)

References

candidate path to improving article

In my opinion, there are two phases to significantly improving this article: 1) whittle down the applications section to just two or three topics, 2) build up that applications section to cover all the applications listed in the article's lead. (The reason I am writing this opinion instead of just taking a shot at it myself is that I am taking on a high risk project. Unless it quickly crashes and burns, I probably will not be doing any more Wikipedia editing for the rest of this year. Perhaps a view of where an edit will fit into a larger editing plan will entice a couple more people to improve this article.)

Concerning the whittling phase, I think the "geometric power series" could be moved into a new Historic insights section on Madhava of Sangamagrama (c. 1340 – c. 1425). And the Repeating decimals section might be a better fit in the Specific geometric series subsection. Concerning the build up phase, the article's lead currently lists these applications: physics, engineering, biology, economics, computer science, queueing theory, and finance. Gj7 (talk) 23:39, 7 June 2021 (UTC)

I moved the "Repeating decimals" subsection, previously in the "Applications" section, into the "See also" section. I think the "See also" section may become a highlight of the article. Gj7 (talk) 22:16, 10 January 2022 (UTC)

Geometric derivation of closed sum formula

Under "Sum" there is a picture that claims to give a geometric proof. I can not make sense of it. From "the first n+1 terms of S_n" the confusion just compounds. It's a shame, because I don't doubt that this could be done properly, and someone seems to have put a lot of effort into trying, but I currently don't see how this can be salvaged. --St.nerol (talk) 23:35, 13 April 2022 (UTC)

I have not checked this talk page for months. Pardon me for my delayed response to your post. The geometric proof in the article definitely requires a different perspective. However, for anyone already familiar with the algebraic proof of the closed form of the geometric series, I think gaining the geometric perspective of that same proof would be rewarding. That geometric proof starts by representing each term in the partial geometric series by the area of an overlapped similar triangle. (That is the top plot.) From an overview, the rest of the proof is removing the overlapped areas, resulting in the area of a large triangle less the area of a small triangle, a geometric interpretation of the closed form equation.

Is there some way to better show that the similar triangles in the top plot are initially overlapped? Perhaps a 3D view? Gj7 (talk) 21:15, 18 April 2022 (UTC)

I added an example to the first step of the geometric proof. If there is still confusion about that first step or the other two steps, let me know. Gj7 (talk) 00:40, 21 April 2022 (UTC)

a simpler Proof of convergence section?

I don't think the alternative algebraic derivation is necessary in the Proof of convergence section. Also, the telescoping series derivation doesn't seem particularly illustrative to me. Therefore I'm thinking about removing the section's alternative algebraic derivation and replacing the telescoping series derivation with a geometric interpretation of convergence: the area of the white triangle in the adjacent figure is the remainder which goes to zero as n increases, provided |r|<1. Any objections to this proposed change? Gj7 (talk) 19:58, 11 May 2022 (UTC)

The changes to the Proof of convergence and Rate of convergence sections ended up shortening the article by about three thousand characters. Gj7 (talk) 14:31, 25 May 2022 (UTC)

geometric power series application?

In the article's "Coefficient a" section, I describe the relationship between the geometric series and the power series. In the article's "Applications" section, there is a "geometric power series" application. Is there a reference for a series named the "geometric power series"? Gj7 (talk) 20:51, 26 December 2020 (UTC)

The comments about differentiating the geometric series have been moved to the Historic insights section on Nicole Oresme. Therefore, all that remains is a description of Gregory's series. In my opinion, there should also be a Historic insights section on Madhava of Sangamagrama and this description of Gregory's series should then be moved there (and the "geometric power series" subsection should then be deleted). Gj7 (talk) 21:49, 7 June 2021 (UTC)

I finally changed this heading name from "geometric power series" to "integration" and I attributed the resulting series to Madhava of Sangamagrama. Gj7 (talk) 17:31, 4 June 2022 (UTC)

Correction for topic 'Mistake in section "Repeating decimals"'

The representation of 7/9 in base 12 is 0.94 (and NOT 0.91). Sorry for the typo, I could not change it after publishing my comment. 2001:4C4E:10FE:8A00:5803:4BB7:58AC:9406 (talk) 11:34, 28 October 2022 (UTC)

Mistake in section "Repeating decimals"

The statement "In fact, any fraction that has an infinitely repeated pattern in base-ten numbers also has an infinitely repeated pattern in numbers written in any other base." is fundamentally wrong.

Infinitely repeating patterns occur when dividing by such a number, which contains a prime factor, that is not part of the base of the number system. That means, finite sequences will occur only for a limited number of divisors (in base 10 only for the primes 2 and 5 (and these powers and combinations; in base 2 only for the prime 2 and its powers). Division by any other primes (or their powers and combinations) will result in infinitely repeating patterns.

The example for 0.777... (which is equal to 7/9) used base 2, where it is also an infinitely repeating binary number, but this is no proof. It is only because base 2 does not include the prime number 3, which is the prime number included in the division (7/3^2). If you consider the same in base 3, 7/9 is written as 0.21, or in base 9 it is 0.7. It will be represented in finite length also in base 12 for example, 0.91 would be the representation. 2001:4C4E:10FE:8A00:5803:4BB7:58AC:9406 (talk) 11:20, 28 October 2022 (UTC)

There is an infinitely repeating pattern of zeros following any finite-length representation. These numbers in fact have two infinitely repeating representations, one with an infinitely repeating pattern of zeros and another with an infinitely repeating pattern of the base minus one. —David Eppstein (talk) 15:38, 28 October 2022 (UTC)

Opening sentence is confusing.

"In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms." We can't sum an infinite number of terms. This needs to be broken down along the lines as follows: "Each term of a geometric series is the sum of a finite number of terms of a geometric sequence" or similar. TorlachRush (talk) 19:50, 12 April 2023 (UTC)