Talk:Geometric series

Original research?

Last year after reading Thousand Brains - a new theory of intelligence by Jeff Hawkins, I started to think about the importance of a useful mental model when learning math. As described in A Thousand Brains, being actionable and simple and hierarchical are characteristics of a useful mental model. Intending to help high school math students, I posted an image of a map of polynomials with the geometric series at the origin, showing the relative locations of the power series, the Laurent series, and matrix polynomials ... all of which are covered in their own Wikipedia articles.

Today, an anonymous editor removed that map of polynomials and claimed it is original research. Can an image that shows a particular organization of existing content be original research? If so, it seems that most images in Wikipedia articles would also be original research (e.g., during the image upload process there is a checkbox to verify that the person uploading the image is also the image creator). Gj7 (talk) 23:03, 25 July 2022 (UTC)

I agree with the anonymous editor. This appears to be original research, only dubiously relevant to the topic, overly promotional for the book, and not sourced by reliable publications in the mathematics literature. I don't think it belongs in this article. —David Eppstein (talk) 23:24, 25 July 2022 (UTC)

The quotes from the book were not intended as a promotion, but as a reference about the task of finding an actionable, simple, and hierarchical mental model is a key to gaining expertise in anything, including math. Why would the geometric series being at the origin of the image be only dubiously relevant to the topic? Do you know of a different and better mental model of math that illustrates the relationship between different series? — Preceding unsigned comment added by Gj7 (talk • contribs) 23:49, 25 July 2022 (UTC)

Though this map had been removed, significant chunks of the remaining text still referenced it. I've now begun removing those chunks as well. RowanElder (talk) 16:30, 27 August 2024 (UTC)

Any objections to restructure?

I recently began editing this page as a newcomer task, copyediting section by section from the early parts without much concern for the overall article structure. However, as I've kept editing chunk by chunk, that overall structure has begun seeming like it needs work.

(1) The examples section has been bizarre. (A) The first examples subsections' texts were structured around a missing figure that had been removed as original research without changing the text. (B) Most of the later examples subsections' texts are excerpts from articles on examples of power series that are not geometric series. (C) The second set of binary representation examples (this was included twice in different sections for some reason) is also mostly about power series that are not geometric series.

(2) In general, the topics of convergence and derivation of the closed form are handled multiple times, sometimes with exactly the same content reiterated in different sections. The sections' names aren't always clearly related to their content; there are "convergence" and "closed form" subsections of the "Components" heading.

In accord with the general advice to "be bold" I'll go ahead and start major-restructuring edits in a couple days, but I'll be happy to discuss the choices here. I intend to (A) reshape the "Components" heading to just include information on a, r, and the terms of the series (B) move convergence and closed form information into the "Sum" heading and (C) delete most "power series" examples and move the rest into a dedicated heading like "Comparison to General Power Series" which preserves the content but adds clearer comparison and motivation for the comparison. For instance, a "matrix exponential" subsection can only make sense to include *after* a subsection on 1 / (I - M) matrices, which also play crucial roles in physics and have much more direct connections to the geometric series. RowanElder (talk) 17:09, 28 August 2024 (UTC)

I began this with (C) today. In the course of that work, I've decided to also (D) add a section on geometric series in higher mathematics. The scattered references to p-adics, Banach spaces, and so on deserve better ordering, and the article would also benefit from additional links to abstract algebra via the ring of formal power series and via generalizations of the base sequences from sequences of real or complex numbers to sequences of elements of rings and semirings. In particular, (D) will include bringing in a proper "matrix geometric series" subsection to make up for deleting the matrix polynomial and matrix exponential from the Examples section. RowanElder (talk) 04:32, 30 August 2024 (UTC)

I began (A) and (B) today. The "Parameters" section will need more work to include important encyclopedic reference material relevant to non-mathematicians about the application and interpretation of geometric series, for instance that "a" defines the units of the expression and several examples of such units. These are big edits so I'm happy to discuss them; I think they're improvements overall but they're certainly not done and I may be accidentally introducing some new errors edit-by-edit. RowanElder (talk) 15:16, 30 August 2024 (UTC)

(A), (B), and (C) now seem settled in shape. I'll start (D) soon. When I do, I plan to place it after the applications section. I may move the applications ahead of the "connections to power series" section as well, since it seems more likely to be relevant to more readers. I also plan to expand the parameters section with some discussion of the parameters' interpretations in applications. RowanElder (talk) 21:52, 31 August 2024 (UTC)

The more I work around the "History" section the more a couple of issues stand out to me.

(1) Zeno is worth citing as an early figure but the section is poor, in particular oversimplifying the contemporary Greek mathematical concerns about the possibility of infinities and misrepresenting the contemporary Greek philosophical applications of the paradoxes.

(2) The Euclid section seems more textbook than encyclopedic.

(3) The history section should almost certainly include subsections on Euler and Cauchy for the history of the formalization of series convergence.

This isn't the sort of work I intend to do on Wikipedia for now, but I hope that some later visitor to this page might help with these issues and might find it useful to have them stated explicitly here now. RowanElder (talk) 16:23, 2 September 2024 (UTC)

I got started on (D) but before going further on it I've been reviewing several of the other Wiki pages that I would be linking to in the section, from fixed point iterations to Fredholm theory. These need work and often lack the citations that I would want to find and bring over to this page, so I'll move more slowly on (D) than I did on (A), (B), and (C).

I therefore consider my vision for the major page restructuring finished at this point. I'll continue to revisit and expand "generalizations beyond real and complex values" and "connection to power series" a little further, for instance definitely including something further on Liouville series in the "power series" and on matrix & operator resolvents and p-adics in the "generalizations," but none of that would be restructuring. RowanElder (talk) 15:13, 4 September 2024 (UTC)

It's surprising and also nice to see that, a few weeks later, page views of this article have apparently risen by ~40% since I did this restructuring. I thought it might just be "beginning of school year" seasonality but the other math pages don't seem to have similar rises. Nice to see this work has a tangible impact! RowanElder (talk) 19:32, 20 September 2024 (UTC)

This was an error: this page is just more seasonal than others. It routinely has this rough size of lift at the beginning of any school year. RowanElder (talk) 04:37, 24 October 2024 (UTC)

In fact, doing these edits was unrewarding, the wikiproject has been unwelcoming, and I now regret spending the time. I have decided to stop contributing to Wikipedia's mathematics for the time being. RowanElder (talk) 19:19, 25 October 2024 (UTC)

Circular Argument within the proof section for the Convergence of the Geometric series

As of right now - the article "proves" the convergence of the geometric series in the "Parameters section" , via the Ratio test - as of right now the Ratio test is proven via the convergence of the geometric series, which leads to a circular argument. I would suggest to provide an alternative proof for the geometric series, for example by considering the limit of the difference between the sum x^k and 1/(1-x) Mpi31415 (talk) 18:17, 1 September 2024 (UTC)

Funny and well caught for that detail. I think it's alright, though. There are already two alternative proofs in the text as it is, in the "Sum" section (an algebraic proof and a geometric proof). At most we need to replace the link to "ratio test" with an internal link down to one of those other proofs; in that case the "Sum" section should probably be given a note that its proof is the basis for the ratio test. RowanElder (talk) 00:34, 2 September 2024 (UTC)

This is done. RowanElder (talk) 16:24, 2 September 2024 (UTC)

First Graphic Has an Error

The first graphic has an error in it where it is labeled 1/8 it should be 1/16. Please fix. — Preceding unsigned comment added by Hkbourne (talk • contribs) 21:50, 4 October 2024 (UTC)

The first graphic should be labeled 1/16 where it is labeled 1/8. Hkbourne (talk) 21:52, 4 October 2024 (UTC)

The labels on the graphic are side lengths of squares: 1/2, 1/4, 1/8, etc. They appear correct to me. —David Eppstein (talk) 23:42, 4 October 2024 (UTC)

Yes, they also look correct to me. RowanElder (talk) 18:39, 5 October 2024 (UTC)

Mathematical analysis or real analysis

I mean, real analysis studies about real numbers and this includes geometric series. So, why mathematical analysis, @Jacobolus? Dedhert.Jr (talk) 16:00, 4 November 2024 (UTC)

You can certainly have a geometric series of complex numbers. But let's go for just "in mathematics". The point of these initial phrases is to orient readers who have no idea what the topic is and accidentally arrive here, not to carefully describe which fields are relevant. –jacobolus (t) 16:02, 4 November 2024 (UTC)

Hmm... That makes sense. Dedhert.Jr (talk) 16:04, 4 November 2024 (UTC)