Editing Talk:Grandi's series

{{dyktalk|12 November|2006|entry=...that '''[[Grandi's series]]''' 1 − 1 + 1 − 1 + · · · is [[divergent series|divergent]] and appears to equal [[0 (number)|0]], yet in some sense "sums" to [[one half|<sup>1</sup>⁄<sub>2</sub>]], producing a paradox once linked to the [[creation (theology)|creation]] ''[[ex nihilo]]'' of the universe?}}
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==Talk==
Should the section on the Cessaro sum say Holder sum instead? I know they're equivalent, but... <small>—The preceding [[Wikipedia:Sign your posts on talk pages|unsigned]] comment was added by [[User:Dchudz|Dchudz]] ([[User talk:Dchudz|talk]] • [[Special:Contributions/Dchudz|contribs]]) 19:00, 10 November 2006  (UTC{{{3|}}})</small>

:Well, my sources call it Cesaro. I'm not familiar with Holder summation, but if it's equivalent then I wouldn't worry about it. [[User:Melchoir|Melchoir]] 21:26, 10 November 2006 (UTC)

Wow, Melchoir, thank you for writing this article! [[User:Mstroeck|mstroeck]] 18:53, 18 November 2006 (UTC)

:Ja! [[User:Melchoir|Melchoir]] 19:13, 18 November 2006 (UTC)

== Sort out references ==

For the record, I'm commenting here that I'm now going to drop some references and their claims, and I'm re-integrating the References section. It's become clear that certain facts are repeated by multiple authors and traceable to primary sources, and some claims are just made up, and now that I've started identifying the difference, there is little point in including the latter. Books that don't claim to be reliable historical sources don't need us to emphasize that point. [[User:Melchoir|Melchoir]] 20:53, 8 December 2006 (UTC)


== Some information ==

Concerning Hölder and Cesàro, for details one is referred to 

* E.W. Hobson, ''The theory of functions of a real variable and the theory of Fourier's series'', Volume II, 2nd edition, reprinted (Cambridge University Press, 1950), section 44.

Historically, Hölder's approach dates from 1882 (''Math. Annalen, Vol. XX, (1882),'' p. 535) and Cesàro's from 1890 (''Bulletin des Sciences Math., Vol. XIV (1890)'', p. 114). It was Knopp (''Grenzwerte von Reihen bei der Annäherung an die Convergenzgrenze'', Inaugural Dissertation, Berlin 1907) who first showed that a series which is summable <math>\ (H,k)</math> (that is summable in accordance with Hölder's definition, of order <math>\ k</math>) is necessarily summable <math>\ (C, k)</math> (that is summable in accordance with  Cesàro's definition, of order <math>\ k</math>). The converse was established by Schnee (''Math. Annalen, Vol. LXVII (1909)'', p. 110) and by Ford (''Amer. Journ. of Math., Vol. XXXII (1910)'', p. 315).

You may take the above information over in your reworking of the present and other related articles.

--BF 02:01, 9 December 2006 (UTC)

:Thanks for the info! In the last few weeks, I've learned some of this, but I didn't know the historical details.
:I'm not sure if this article should talk about the higher-order methods, but they should definitely appear in the more dedicated articles.
:One thing worries me: what exactly did Hölder propose in 1882? One or two of my sources claim that the 1890 Cesàro sum was the first systematic method of summation beyond the convergent one. It's implied that theorems by Abel and Frobenius that we now think of as relating different summation methods weren't posed as such at the time, and early efforts by Leibniz and Euler don't count because they weren't stated in precise terms. So it's conceivable to me that Hölder's work was along the same lines; on the other hand, it could be a scoop of Cesàro. Anyone know? [[User:Melchoir|Melchoir]] 02:22, 9 December 2006 (UTC)


You are welcome. I would recommend you to consult the book by Hobson cited above (Dover has reprinted this book in 1957). 

Briefly, '''the Hölder and Cesàro techniques are in principle distinct'''; it was not until 1909 that one knew that a series is summable <math>\ (H, k)</math> if and only if it is summable <math>\ (C, k)</math> and not until 1913 that one knew that the two methods are '''equivalent'''. 

For completeness, let <math>a_1 + a_2 + \dots</math> be the series to be considered. Let <math>s_n = a_1 + \dots + a_n, n=1,2,\dots</math>. The Hölder partial sums are <math>\{ h_n^{(k)}\} </math> where <math>h_n^{(1)} = (s_1+\dots+s_n)/n, </math> <math>h_n^{(k)} = (h_1^{(k-1)} + \dots + h_n^{(k-1)})/n, k=2,3,\dots.</math> The Cesàro partial sums are <math>\{ C_n^{(k)} \}</math> where <math>C_n^{(k)} = [k! (n-1)!/(n+k-1)!] s_n^{(k)}</math>
in which <math>s_n^{(0)} = s_n, </math> <math> s_n^{(1)} = s_1^{(0)} + \dots + s_n^{(0)}, </math> <math>s_n^{(k)} = s_1^{(k-1)} + \dots + s_n^{(k-1)}, k=2,3,\dots .</math> The above series is summable <math>\ (H,k)</math> if <math>\lim_{n\to\infty} h_n^{(k)}</math> exists. It is summable <math>\ (C, k)</math> if <math>\lim_{n\to\infty} C_n^{(k)}</math> exists. It can be shown that when <math>s = \lim_{n\to\infty} s_n</math> exists, both <math>\lim_{n\to\infty} h_n^{(k)}</math> and <math>\lim_{n\to\infty} C_n^{(k)}</math> indeed exist for all <math>\ k</math> and are equal to <math>\ s</math>. Hobson refers to <math> \lim_{n\to\infty} s_n</math> as '''ordinary''' sum and to <math>\lim_{n\to\infty} h_n^{(k)}</math> and <math>\lim_{n\to\infty} C_n^{(k)}</math> as '''conventional''' sums, which satisfy the ''consistency condition'', that they yield the same result as the ordinary sum when applied to convergent series. 

The proof for the '''complete equivalence''' of the two methods of summation (or of summability of order <math>\ k</math>) is given by Schur (''Math. Annalen, Vol. LXXIV (1913)'', p. 447) and by Hahn (''Monatshefte f. Math. u. Physik, Vol. XXXIII (1923)'', p. 135). For details (including the proof of the equivalence of the two methods) see sections 55-57 of the above-cited book by Hobson. 

I must admit that there is some confusion in textbooks concerning the two methods. For instance, Whittaker and Watson (4th edition, reprinted, Cambridge University Press, 1962, Sec. 8.43) describe the Cesàro method in exactly the same way as Hobson describes the Hölder method; there is no reference to Hölder's method in the book by Whittaker and Watson. I believe that on this matter Hobson should be considered as the authority. 

--BF 15:13, 9 December 2006 (UTC)

:Huh. Actually, yes, I was under the apparently mistaken impression that the Cesàro methods were the iterated ones. By pure luck, it doesn't matter for this article, but if there's confusion to be cleared up then we should do it somewhere. You seem to have good references in front of you; could you be persuaded to start up [[Hölder summation]] and/or expand [[Cesàro summation]] with information on history and higher-order definitions? [[User:Melchoir|Melchoir]] 23:14, 9 December 2006 (UTC)


Thank you for your suggestion. Expansion of the article on the [[Cesàro summation]] technique is problematic, since one will have to ascribe what at present is ascribed to Cesàro to its rightful author Hölder. This, I believe, should be in the first place the responsibility of the person or persons who have initiated the article on the subject, not least for the fact that this or these persons should have the opportunity to defend their standpoint as expressed in their article. This is why I initiated the present Talk. 

As regards starting up an article on the [[Hölder summation]] method, it seems that this requires, roughly speaking, only renaming the article [[Cesàro summation]] as [[Hölder summation]]. As the two articles are necessarily closely related, I prefer that the entire task be undertaken by the initiator(s) of [[Cesàro summation]].

If you wish to modify the article on [[Cesàro summation]] and further to create a page on [[Hölder summation]], please feel free to do so; in doing so, please also feel free and take over whatever you feel necessary from my earlier texts on this page. In such case, I shall check the two articles and possibly amend them if necessary. In the event that you decide to work on these two articles, please indicate the confusion that exists in the literature of the subject concerning the two summation methods (even despite the fact that they are mathematically equivalent).

--BF 16:12, 10 December 2006 (UTC)

:Sorry, I don't think I'm going to work on those articles myself. FYI, [[Cesàro summation]] was written largely by [[User:Merge]], who doesn't seem to be around much. This particular talk page ([[Talk:Grandi's series]]) is hard to find and seldom read except by me, so if you want to really raise the issue, I encourage you to leave a message at [[User talk:Merge]], [[Talk:Cesàro summation]], or [[Wikipedia talk:WikiProject Mathematics]]. Or you can just wait until I make this a Featured Article, at which point lots of people will visit this talk page, but there's no guarantee when that'll happen! [[User:Melchoir|Melchoir]] 18:24, 10 December 2006 (UTC)


Thank you for your recommendations.

--BF 19:21, 10 December 2006 (UTC)

== Merger? ==

Is iit really necessary to have 5 different articles about Grandi's series? I'm looking at https://en.wikipedia.org/wiki/Grandi%27s_series , https://en.wikipedia.org/wiki/History_of_Grandi%27s_series , https://en.wikipedia.org/wiki/Grandi%27s_series_in_education , https://en.wikipedia.org/wiki/Summation_of_Grandi%27s_series , and https://en.wikipedia.org/wiki/Occurrences_of_Grandi%27s_series . Thoe last three don't even have talk pages. I volunteer to help in the merger, but I need permission first. So, a vote?

== Removal of "proof"s ==

I removed two "proof"s that are nonsensical: the first one using indefinite integrals is invalid because indefinite integrals are only defined up to addition/subtraction of constants, while the second one using a geometric series basically duplicates, in a bad way, what was already given.

In general, I think this article needs a lot more love. There are rigorous methods of assigning a value to this series, and I agree with the above suggestion that the article on summation be merged here.--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 02:33, 30 October 2016 (UTC)

== So what is the value? ==

After reading the article, I am still no wiser at the value of this series. So what exactly is its value? [[Special:Contributions/2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91|2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91]] ([[User talk:2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91|talk]]) 01:39, 13 August 2018 (UTC)
:As the article [https://en.wikipedia.org/w/index.php?title=Grandi%27s_series&oldid=848412274 already said] back in July 2018, it lacks a sum in the usual sense, but if you want to generalise summation so that it produces a value for this series, the only value that makes sense is 1/2. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 02:56, 18 May 2019 (UTC)
::Would 0.5±0.5 not encompass it better?[[User:Argentum Kurodil|Argentum Kurodil]] ([[User talk:Argentum Kurodil|talk]]) 10:39, 22 April 2020 (UTC)

== Euler quote? ==

The article seems to attribute an unlikely quote to Euler: "...Assume everything is okay, and if the arrived-at solution works, you were probably right...". I think this is someone else (Lehman?) paraphrasing Euler. Perhaps that could be clarified. [[User:Pburka|pburka]] ([[User talk:Pburka|talk]]) 03:58, 1 February 2025 (UTC)

== Proposed merge of [[Summation of Grandi's series]] into [[Grandi's series]] ==

The article "Summation ..." is an unmaintained [[WP:REDUNDANTFORK]]; to the extent the content is encyclopedic and supported by reliable sources, it should just be included in the main article [[Grandi's series]]. --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 21:04, 25 January 2026 (UTC)

:'''Support''' merger, this doesn't serve any separate purpose. It might also be worth looking at [[History of Grandi's series]] and [[Occurrences of Grandi's series]], although these might have slightly stronger justifications of being standalone articles. [[User:Thebiguglyalien|<span style="color:#487d30">Thebiguglyalien</span>]] ([[User talk:Thebiguglyalien|<span style="color:#714e2a">talk</span>]]) [[Special:Contributions/Thebiguglyalien|🛸]] 02:55, 29 January 2026 (UTC)
::Yikes.  I would support also merging [[Occurrences of Grandi's series]], which does not seem to have a clear case for a standalone article.  [[History of Grandi's series]] seems like a more complicated case; for one thing, it seems to have more content and more sources than this article ([[Grandi's series]]). --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 19:42, 29 January 2026 (UTC)
:'''Support''' merge of [[Summation of Grandi's series]] and [[Occurrences of Grandi's series]] into [[Grandi's series]]. I'm currently undecided about [[History of Grandi's series|the "History" sub-article]]. It's longer, but parts may be redundant, and some of it reads like [[WP:NOR|OR]]: {{tq|Presumably he arrived at this series by repeated substitution}}, etc. Cleanup would be easier if we consolidate the other sub-articles first. [[User:Stepwise Continuous Dysfunction|Stepwise Continuous Dysfunction]] ([[User talk:Stepwise Continuous Dysfunction|talk]]) 21:11, 30 January 2026 (UTC)

:'''Ambivalent''' For what it's worth, I originally created the sub-articles, but a lot of their mental context has left me. Also, I haven't been maintaining them closely, so if some OR has seeped in, I wouldn't necessarily know about it. My intuition is that the sub-articles already have enough sources that they ''could'' be written in such a way to justify being left as standalone. But in practice, it's arguable that Summation and Occurrences don't live up to that. It's also worth noting that the main article doesn't have a fleshed-out summary of either of those articles, so it's not as if a merge would necessarily sacrifice any existing prose. [[User:Melchoir|Melchoir]] ([[User talk:Melchoir|talk]]) 18:51, 4 February 2026 (UTC)