Talk:Cesàro summation

message: "...Cesàro summation is an alternative means of assigning a sum...."  Ha ha, "means".  I hope the pun was intentional. 128.255.45.80 (talk) 23:43, 21 March 2010 (UTC)
thread_title: Pun

message: The article currently says that the generalized Cesàro sum can be restated as 
:(C,\alpha)-\sum_{j=0}^\infty a_j = \lim_{n\to\infty} \sum_{j=0}^n \frac a_j.
However, the article also says that if the series converges then the Cesàro sum is equal to the limit, and that the generalized Cesàro sum is just an iterated version of the normal Cesàro sum – this would seem to imply that if \sum_{j=0}^\infty a_j exists then the difference on the left-hand side should be zero. Am I missing something? Joriki (talk) 21:15, 26 November 2010 (UTC)
replies: text: The LHS is not a difference.  It's (bad but standard) notation for the sum. Sławomir Biały (talk) 12:24, 4 February 2011 (UTC)
replies: text: Currently, the ASCII hyphen is being automatically converted to a minus sign because it's inside <math> tags.  Perhaps this would be less confusing if we were to use a non-math hyphen or an en-dash or something?  Joule36e5 (talk) 23:54, 1 February 2013 (UTC)
replies: text: I changed it to an en dash.  Joule36e5 (talk) 09:20, 26 February 2015 (UTC)
thread_title:  difference between Cesàro sum and limit? 

message: "Since a sequence that is ultimately monotonic either converges or diverges to infinity, it follows that a series which is not convergent but Cesàro summable oscillates."
Note that it doesn't have to be a regular cycle...  For example:
n = {1,2,3,4,5,6,7,8,...}   integers greater than 0
x_n = cos(n)+cos(n sqrt(2))  cycle that never repeats because of impossible factoring
This is a trivial example.  Since by definition, a transcendental number never repeats, this means that it is impossible to have harmonics with n and n sqrt(2).  Since the natural period of the cosine function is a multiple of Pi, some care has to be taken when constructing such examples to not allow the terms inside the cosine functions have certain values (or patterns).  n and n*1.41... with n being positive integers satisfies this requirement.
71.196.246.113 (talk) 09:56, 19 January 2012 (UTC)
replies: text: Sorry, meant to mention my question.  Does anyone think this is important enough of a point to add to the article.  And I dare someone to prove that the square root of 2 is rational!  ;)  (It's an old disproof in any decent book on proofs)71.196.246.113 (talk) 10:01, 19 January 2012 (UTC)
thread_title:  Irregular oscillator 

message: I do read quite often exactly what I read here about the cesàro mean of the Grandi series, yet I never found a plausible explanation. This article states the following:
:\lim_{n\to\infty} t_n = 1/2.
with  t_n = \frac{1}{n}\sum_{k=1}^n s_k 
From my understanding (and the German Wiki Article) however, this is false.
Starting from the limit: \lim_{n\to\infty} t_n I think it should become:
:\lim_{n\to\infty} t_n = \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n s_k. = 0
Dreikommavierzehn (talk) 22:14, 12 May 2015 (UTC)
replies: text: The German article :de:Cesàro-Mittel () is incorrect. For more details on the Cesàro sum of Grandi's series, including a citation, see Summation of Grandi's series#Cesàro sum. Melchoir (talk) 23:12, 12 May 2015 (UTC)
replies: text: The explanation from the link makes sense, yet I don't see the correctness in the above example. The limit of 1/n * something will still be 0. At the very least, I think that the statement written with a limit is flawed. 2001:628:2010:22:14DB:2B64:CC76:577A (talk) 08:42, 13 May 2015 (UTC)
replies: text: Yes, if the series \sum_{k=1}^\infty s_k = \lim_{n\to\infty}\sum_{k=1}^n s_k converged, then the limit \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n s_k would be 0. That's the sense in which the limit of 1/n * something is 0. However, in this case, the "something" by itself, \sum_{k=1}^\infty s_k does not converge. It diverges to positive infinity, and \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n s_k = 1/2 measures how quickly it does so. Melchoir (talk) 20:06, 13 May 2015 (UTC)
replies: text: Alright. But now it looks like \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n s_k =  "0∗∞" an indeterminate form. Aside from the arguments provided by Summation of Grandi's series#Cesàro sum, I'm under the impression that the argumentation always boils down to some form of reordering the partial sums. Which clearly cannot be done. Dreikommavierzehn (talk) 22:22, 14 May 2015 (UTC)
replies: text: That's not how indeterminate forms work. I don't wish to sound rude, but if you're looking for a fuller explanation, you might get a better response at Wikipedia:Reference desk/Mathematics. Melchoir (talk) 03:52, 15 May 2015 (UTC)
replies: text:  Thank you, I'll hit my books. Dreikommavierzehn (talk) 11:02, 17 May 2015 (UTC)
thread_title:  Example of the Grandi series 

message: 1. Version 26 June and 10 Oct. of the article has in section 'Definition' (without symbolic notation, and added underlining):      (A) "A series is called Cesàro summable, ..., if the average value of its partial sums tends to ... .";  version 8 July and 11 Sept. has:      (B) "A sequence is called Cesàro summable, if the arithmetic mean of its partial sums tends to a limit";  and Sławomir Biały mentions as definition (above, 10 Oct.):      (C) "A sequence is Cesaro summable if the sequence of partial sums has a mean that converges".  It's not at all clear why (B) should be wrong, while (C) should be right.
2. Sławomir Biały restores a version with (last sentence before 'Exemples') the clearly incorrect:  "For any convergent sequence, the corresponding series is Cesàro summable".
Conclusion: no valid arguments for restoring version 26 June 2017 are presented. -- Hesselp (talk) 17:54, 12 October 2017 (UTC)
replies: text: I have removed the incorrect sentence.  Do not restore your incorrect revision of the article.  Sławomir Biały (talk) 23:24, 12 October 2017 (UTC)
replies: text: I agree with the removal of the incorrect  "For any convergent sequence, ...".  Can you, Sławomir Biały, specify why my wording  (B) "A sequence is called Cesàro summable, if the arithmetic mean of its partial sums tends to a limit"  is incorrect, while your wording   (C) "A sequence is Cesaro summable if the sequence of partial sums has a mean that converges"  should be correct?  Moreover, can I assume consensus on stating that the definition of  the Cesàro-sum of a infinite sequence  is identical with the definition of  the Cesàro-sum of a infinite series  and as well of  the Cesàro-sum of a infinite sum ?  Who can mention differences? -- Hesselp (talk) 10:34, 13 October 2017 (UTC)
replies: text:  I am not interested in engaging in discussion with you.  It is not required that I explain why your revision is incorrect.  Sławomir Biały (talk) 10:46, 13 October 2017 (UTC)
replies: text: To watchers of this page:   Who has argumented objections against version 11 September 2017 of the article with the definition of 'Cesàro summable' as mentioned by Sławomir Biały 10 October 2017 ? (Considerable shortening of section 'Definition', without loss of information.) -- Hesselp (talk) 22:25, 14 October 2017 (UTC)
replies: text:  I oppose the proposed revision, because the definition of Cesaro summability is incorrect.  Sławomir Biały (talk) 03:59, 15 October 2017 (UTC)
replies: text: On 10 October 2017 you (Sławomir Biały) wrote: "A sequence is Cesaro summable if the sequence of partial sums has a mean that converges".  What do you see now is as being incorrect in that wording of 'the definition' ?  Without your answer the proposal for a revision (shortening) will remain unchanged. -- Hesselp (talk) 09:03, 15 October 2017 (UTC)
replies: text:  I do not object to the correct definition in my post on this discussion page and the current stable revision of the article.  I do object to the incorrect one.  Suggested edits should be based on the current stable revision, not on a manifestly incorrect revision of the article.  Sławomir Biały (talk) 11:20, 15 October 2017 (UTC)

text: I missed all the drama, but the current version is fine.  I don't think there are any major changes needed.  Hesselp, the version you had in place in September before it was restored was incorrect.  --Deacon Vorbis (talk) 14:54, 15 October 2017 (UTC)
replies: text: To Deacon Vorbis.  I can see your changes in the second part of the article as improvements. As I see the deletion of the sentence  "For any convergent sequence, the corresponding series is Cesàro summable" in the Definition section on 12 October 2017. (See my proposal 30 June 2017).  According to me, the readability of the Definition section can be improved by the much shorter wording (without changing the content) as in my edits. I shouldn't call that a 'major change'.  But by what reason do you write that this edits were incorrect?    Please be specific. -- Hesselp (talk) 19:31, 15 October 2017 (UTC)
replies: text:  Your "much shorter wording" changed the meaning completely.  Sławomir Biały (talk) 19:39, 15 October 2017 (UTC)
thread_title:  On restoring 10 October 2017 version 26 June 2017 

message: I get the impression that in the German language the terms "Folge" for sequence and "Reihe" for series are discriminated in a more rigorous way than in English, and the efforts to emphasize their differences are more exerted. Certainly, one can discard differences, call them nitpicking, and neglect them, but some see these differences as useful, and I certainly cannot discuss the English idiomatic.
However, I am afraid that a certain scent around WP might come from several dead horses lying around, because sticks, driving them home were dropped by editors, feeling threatened by getting blocked.
I know about WP just reflecting the (previous?) state of the art in reliable sources. Purgy (talk) 08:04, 17 October 2017 (UTC)
thread_title: Remarks about "Folge" and "Reihe" in German