Talk:Grandi's series

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Should the section on the Cessaro sum say Holder sum instead? I know they're equivalent, but... —The preceding unsigned comment was added by Dchudz (talk • contribs) 19:00, 10 November 2006 (UTC)

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. Melchoir 21:26, 10 November 2006 (UTC)

Wow, Melchoir, thank you for writing this article! mstroeck 18:53, 18 November 2006 (UTC)

Ja! Melchoir 19:13, 18 November 2006 (UTC)

[edit] 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. Melchoir 20:53, 8 December 2006 (UTC)


[edit] 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 \ (H,k) (that is summable in accordance with Hölder's definition, of order \ k) is necessarily summable \ (C, k) (that is summable in accordance with Cesàro's definition, of order \ k). 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? 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 \ (H, k) if and only if it is summable \ (C, k) and not until 1913 that one knew that the two methods are equivalent.

For completeness, let a_1 + a_2 + \dots be the series to be considered. Let s_n = a_1 + \dots + a_n, n=1,2,\dots. The Hölder partial sums are \{ h_n^{(k)}\} where h_n^{(1)} = (s_1+\dots+s_n)/n, h_n^{(k)} = (h_1^{(k-1)} + \dots + h_n^{(k-1)})/n, k=2,3,\dots. The Cesàro partial sums are \{ C_n^{(k)} \} where C_n^{(k)} = [k! (n-1)!/(n+k-1)!] s_n^{(k)} in which s_n^{(0)} = s_n, s_n^{(1)} = s_1^{(0)} + \dots + s_n^{(0)}, s_n^{(k)} = s_1^{(k-1)} + \dots + s_n^{(k-1)}, k=2,3,\dots . The above series is summable \ (H,k) if \lim_{n\to\infty} h_n^{(k)} exists. It is summable \ (C, k) if \lim_{n\to\infty} C_n^{(k)} exists. It can be shown that when s = \lim_{n\to\infty} s_n exists, both \lim_{n\to\infty} h_n^{(k)} and \lim_{n\to\infty} C_n^{(k)} indeed exist for all \ k and are equal to \ s. Hobson refers to \lim_{n\to\infty} s_n as ordinary sum and to \lim_{n\to\infty} h_n^{(k)} and \lim_{n\to\infty} C_n^{(k)} 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 \ k) 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? 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! Melchoir 18:24, 10 December 2006 (UTC)


Thank you for your recommendations.

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

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