Talk:Cesàro summation

Sentence doesn't make sense
The following sentence in the article doesn't make sense: "In fact, any series which diverges to (positive or negative) infinity the Cesàro method also leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable."

Should it say something along the lines of "In fact, any series whose partial sums diverge to plus or minus infinity also leads to a sequence that diverges likewise in the Cesàro sense."?

Pun
"...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)

difference between Cesàro sum and limit?
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)
 * The LHS is not a difference. It's (bad but standard) notation for the sum.  Sławomir Biały  (talk) 12:24, 4 February 2011 (UTC)
 * Currently, the ASCII hyphen is being automatically converted to a minus sign because it's inside &lt;math&gt; 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)
 * I changed it to an en dash. Joule36e5 (talk) 09:20, 26 February 2015 (UTC)

Irregular oscillator
"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)
 * 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)

Example of the Grandi series
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)


 * 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. Melchoir (talk) 23:12, 12 May 2015 (UTC)


 * 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)


 * 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)


 * 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, 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)


 * 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 Reference desk/Mathematics. Melchoir (talk) 03:52, 15 May 2015 (UTC)


 * Thank you, I'll hit my books. Dreikommavierzehn (talk) 11:02, 17 May 2015 (UTC)

The series corresponding with a given sequence?
Usually, the form $$\sum_{i=1}^\infty a_i$$ is understood to denote the series corresponding with sequence ${a_{i}}$, not the series corresponding with the sum sequence  ${a_{1}+···+a_{i}}$  of  ${a_{i}}$. So in the article the sentence: "For any convergent sequence, the corresponding series is Cesàro summable ..." is not correct. I propose to replace what has been here until 05:05, 16 July 2016:  "However, as the first example below demonstrates, there are series that diverge but are nonetheless Cesàro summable."

Second proposal: The present text of section 'Definition' can be formulated shorter without loss of information, as:

Definitions A sequence ${a_{i}}$ (and the corresponding series) is called Cesàro summable, if the arithmetic mean of its first $n$ partial sums, $$\frac{1}{n}\sum_{i=1}^n (a_1+\cdots+a_i)$$,   for $n→∞$  tends to a limit  (the Cesàro sum of the sequence and the corresponding series). It is easy to show that any summable sequence (and series) is Cesàro summable, and its usual sum equals the Cesàro sum. However, as the first example below demonstrates, Cesàro summability does not imply usual summability. -- Hesselp (talk) 13:40, 30 June 2017 (UTC)


 * This is not the definition of Cesaro summability. A sequence is Cesaro summable if the sequence of partial sums has a mean that converges (prior to your edits to the article).  I will not engage further with you except to point out this basic mathematical error.   Sławomir Biały  (talk) 00:29, 10 October 2017 (UTC)

On changes 8 September 2017
@ 89.66.65.113. Thank you for discovering and repairing two missing denominators - my fault. But I don't agree with your 'more strict formula': it's just wrong. In case you prefer (the longer):  $$(s_1+\cdots+s_n)/n$$  where  $$s_i = \sum_{j=1}^i a_j$$, please show your arguments on this Talk page. -- Hesselp (talk) 13:24, 8 September 2017 (UTC)


 * IMO there description by words and by formulas diverged a bit. $$ \frac{1}{n} \sum_{i=1}^{n} a_{1}+\cdots +a_{i} $$ is arithmetic mean of first $$n$$ terms. it also equals $$ \frac{s_n}{n} $$. The sequence in example below the formula is made of $$ \frac{s_n}{n} $$. This differs from "arithmetic mean of its first n partial sums" (which should be written as $$(s_1+\cdots+s_n)/n$$). I felt confused when I read the article first time, then I looked at other wikis (Polish, Italian, ...) and made my change. Unfortunately I don't have any math books at hand to check how Cesàro summation is defined in the sources (apparently | Bronshtein and Semendyayev are focused on more "practical" things). — Preceding unsigned comment added by 89.66.65.113 (talk) 19:04, 12 September 2017 (UTC)


 * @89.66.65.113. Can you see the following seven 'equal signs' as being correct?:


 * The arithmetic mean of the first $n$ terms of sequence ${a_{i}}$  =    $$(a_1+\cdots+a_n)/n$$   =   $$\frac{1}{n}\sum_{i=1}^n  a_i$$   =   $$ \frac{s_n}{n} $$.


 * The arithmetic mean of the first $n$ partial sums of sequence ${a_{i}}$  = the arithmetic mean of the first $n$ terms of sequence ${s_{i}}$   = $$(s_1+\cdots+s_n)/n$$   =    $$\frac{1}{n}\sum_{i=1}^n  s_i$$   =   $$\frac{1}{n}\sum_{i=1}^n  (a_1+\cdots+a_i)$$ . -- Hesselp (talk) 20:24, 12 September 2017 (UTC)


 * Now I see how mixing two summation notations ($$\sum$$ and $$a_1+\cdots +a_i$$) confused me. Thank you for patience in clarifying this. While big part of my previous comment is indeed wrong, I still think that splitting the formula to have partial sums explicitly defined is more clear and easier to understand. In case you prefer compactness, formula with both sums written in same convention: $$\frac{1}{n}\sum_{i=1}^n \sum_{j=1}^i a_j$$ also seems to be appealing.  — Preceding unsigned comment added by 89.66.65.113 (talk) 19:30, 13 September 2017 (UTC)


 * My choice for    $$\frac{1}{n}\sum_{i=1}^n  (a_1+\cdots+a_i)$$  (1)    instead of     $$(s_1+\cdots+s_n)/n$$  where  $$s_i = \sum_{j=1}^i a_j$$  (2) had to do with the fact that (1) uses the least number of different letters for variables (a, n, i). We can have different views on your "more clear and easier to understand".   Even more alternatives are: $$(s_1+\cdots+s_n)/n$$  where  $$s_i = a_1+\cdots+a_i$$  (3)   and $$[a_1 + (a_1+a_2) + \cdots + (a_1+\cdots+a_n)] / n $$ (4, only two different letters).


 * You can read german? If so, see for more on the 'best' presentation of this subject: here and here and here. Greetings. -- Hesselp (talk) 11:28, 14 September 2017 (UTC)


 * I like these german pages – there is no mixing of $$\sum$$ with $$+\cdots+$$ in one formula there :-) IMO the most easy explanations are using $$s_i$$ explicitly. My 2nd favourite would be $$\frac{1}{n}\sum_{i=1}^n \sum_{j=1}^i a_j$$, third would be something like $$\frac{1}{n}\sum_{i=1}^n (n-i+1) a_i$$ (single sum, also only 3 symbols, may be considered as an alternate view). As single point is not enough for statistics, it would be good to gather some more opinions on what is the "most efficient" formula to present. I don't intend to edit the article. Best Regards. 89.66.65.113 (talk) 18:48, 14 September 2017 (UTC)

I do not engage with User:Hesselp. He is wrong and does not understand basic mathematics, and instead seeks to waste time with endless discussion. I will revert his further edits to this article. Sławomir Biały (talk) 00:27, 10 October 2017 (UTC)

On restoring 10 October 2017 version 26 June 2017
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)


 * I have removed the incorrect sentence. Do not restore your incorrect revision of the article.   Sławomir Biały  (talk) 23:24, 12 October 2017 (UTC)


 * 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)


 * 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)


 * 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)


 * I oppose the proposed revision, because the definition of Cesaro summability is incorrect.  Sławomir Biały  (talk) 03:59, 15 October 2017 (UTC)


 * 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)


 * 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)


 * 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)


 * 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)


 * Your "much shorter wording" changed the meaning completely.  Sławomir Biały  (talk) 19:39, 15 October 2017 (UTC)

On the choice between the names "Cesàro summable sequence" and "Cesàro summable series"
Comparing the numbers of Google hits results in: "Cesàro summable sequence" 973; "Cesàro summable sequences" 593;  "Cesàro summable series" (sing.+pl.): 690. For all kinds of summability together, Google produces:  "summable sequence" 27 100;  "summable sequences" 47 300;  "summable series" (sing.+pl.): 21 100.

Moreover, the WP article 'Series (mathematics)' says about the modern meaning of the word 'series': The series defined by an infinite sequence of addible terms is the operation of adding this terms one after the other. It's much more difficult, imo, to see 'Cesàro summability' as a property of that operation, than as a property of a sequence (with addible terms).

Who can show arguments in favour of using the name "Cesàro summable series" in the article? Mentioning both names is a possibility as well, of course. -- Hesselp (talk) 16:14, 16 October 2017 (UTC)


 * Please read WP:IDHT and WP:DEADHORSE --Deacon Vorbis (talk) 16:34, 16 October 2017 (UTC)
 * But to add a little, the short version is that the way you give the definition isn't as clear, and your insistence on "sequence" instead of "series" isn't standard. Lest you risk a wider topic ban, it might be a good idea to drop this one.  --Deacon Vorbis (talk) 17:05, 16 October 2017 (UTC)


 * Deacon Vorbis, thank you for (what I see as) a specification of your 'incorrect' (15 Oct.). On: "the definition isn't as clear": Is the following variant a sufficient improvement? Definition:   A sequence/series (with terms $$a_i$$ and partial sums $$a_1+\cdots+a_i = s_i$$)  is called Cesàro summable, if the arithmetic mean of its first $$n$$ partial sums, $$(s_1+\cdots+s_n)/n$$,  for increasing $$n$$ tends to a limit  (the Cesàro sum of the sequence).  On "sequence instead of series isn't standard":  There are two different situations. Just as anyone else, I see that "convergent series" is widely more used than "summable sequence".  But the Cesàro-case is different. See the Google statistics. And on the German page "Diskussion:Cesàro-Mittel"  user HilberTraum states (5 July 2017):   Bisher war mir ein Begriff wie „summierbare Reihe“ ebenfalls unbekannt. Es gibt zwar ein paar Google-Treffer, aber üblich scheint es mir, nur „summierbare Folge“ oder (wesentlich häufiger, weil allgemeiner) „summierbare Familie“ zu definieren. [I think it is usual only to define 'summable sequence' or 'summable family'.] Finally: as far as I know, this Talk Page is not meant for a reaction on remarks concerning the risk of bans. -- Hesselp (talk) 00:21, 17 October 2017 (UTC)


 * "Methods for the summation of series" on EoM:"Cesàro summation methods". Boris Tsirelson (talk) 20:29, 16 October 2017 (UTC)


 * It's much more difficult, imo, to see 'Cesàro summability' as a property of a sequence-sumsequence-pair, than as a property of a sequence (with addible terms). -- Hesselp (talk) 00:21, 17 October 2017 (UTC)


 * Zygmund's Trigonometric series uses both.  Sławomir Biały  (talk) 21:31, 16 October 2017 (UTC)


 * Maybe this idea can be easily realized as well in the WP-article. The wording of the definition remains the same for: A series (with terms ai) is called Cesàro summable if ...,  and for A sequence (with terms ai) is called Cesàro summable if ... . -- Hesselp (talk) 00:21, 17 October 2017 (UTC)


 * If $$(a_1,a_2,a_3,\ldots)$$ is a sequence, the following sentences are equivalent and non-ambiguous, even if some authors consider some of them as improper: "the series $$(a_1+a_2+a_3+\cdots)$$ is convergent", "the series $$(a_1+a_2+a_3+\cdots)$$ is summable", and "the sequence $$(a_1,a_2,a_3,\ldots)$$ is summable". They must not be confused with "the sequence $$(a_1,a_2,a_3,\ldots)$$ is convergent". Although Summable sequence redirects to Series (mathematics), the term "summable sequence" was not defined there. I have just fixed this. A similar edit could be done here, if the phrase "Cesàro summable sequence" is commonly used, which, I believe, is not the case. D.Lazard (talk) 09:42, 17 October 2017 (UTC)


 * @D.Lazard. Do you accept as belonging to your (imo correct) set of equivalent and non-ambiguous sentences as well:  "the infinite series with terms $$a_i$$ is convergent",   "the infinite series with terms $$a_i$$ is summable"  and  "the infinite sequence with terms $$a_i$$ is summable" ? Secondly, you don't see the number (and the contexts) of google hits on "Cesàro summable sequence" and "Cesàro summable sequences" (973+593) as convincing enough for mentioning in Cesàro summation ? -- Hesselp (talk) 12:23, 17 October 2017 (UTC)


 * I don't think it is worth arguing about. WP:DEADHORSE applies.  But picking the Google web hits seems like cherry-picking.   Both Google books and Google scholar show that the research literature prefers "Cesaro summable series" by about a 2:1 ratio.   Sławomir Biały  (talk) 12:48, 17 October 2017 (UTC)


 * Cherrypicking? From Google Books plus Google scholar I get: "Cesaro summable series":   128 + 47 = 175 "Cesaro summable sequences":   106 + 138 = 244 "Cesaro summable sequence":   7 + 31 = 38.     244 + 38 = 282 Did you, Sławomir Biały, use other sources than I ? -- Hesselp (talk) 16:05, 17 October 2017 (UTC)


 * Surely, other sources. I just try the first "scholar" link of Hesselp, and see "Ongeveer 47 resultaten (0,02 sec)"; this is scholar.google.nl (Neterlands?) which could make a difference. Boris Tsirelson (talk) 17:02, 17 October 2017 (UTC)
 * No, I am wrong. My own search gives just 47 results, too. Also I confirm the 138 and 31.Boris Tsirelson (talk)
 * But for Books I get 141 (not 128), 127 (not 106), and 46 (not 7). Maybe different options, preferences, histories...? Yes, "When signed in, you can receive more relevant results and recommendations based on your search activity." Boris Tsirelson (talk) 17:14, 17 October 2017 (UTC)
 * Wonders never cease! I set (on Google Books) 100 results per page (rather than 20), and got completely new numbers: 23, 35, 19 (not 141, 127, 46). What does it mean? Boris Tsirelson (talk) 17:32, 17 October 2017 (UTC)
 * The numbers are comparable, in any case, and it's ridiculous to attempt to legislate what the article should call the thing based on raw Google numbers.  Sławomir Biały  (talk) 17:48, 17 October 2017 (UTC)
 * (edit conflict) Hmmm... For "cesaro summable series", Google Books gives me 23 results always. In the 10 results per page mode it gives 10+10+3 BUT says on page 1 "About 141 results", on page 2 "Page 2 of about 132 results", and on page 3 "Page 3 of 23 results". In the 100 pages per page mode it gives just 23 results, and says so. I have no idea, what does the 141 mean. Results are the same (and in the same order), not just the same number. Boris Tsirelson (talk) 17:53, 17 October 2017 (UTC)


 * Thank you, Tsirel. The following data are not new now any more, but maybe it can be practical to see them in the same configuration as before.  Now with sources  https://www.google.com/search?tbm=bks&q=%22   and   https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=%22 "Cesaro summable series":   136 + 47 = 183 "Cesaro summable sequences":   127 + 138 = 265 "Cesaro summable sequence":   46 + 31 = 77.       265 + 77 = 337 My conclusion out of this data: in the article should be mentioned that "Cesàro summable sequence" and "Cesàro summable series" are used in practice. With exactly the same definition (or with equivalent definitions), and therefore with the same meaning. -- Hesselp (talk) 18:52, 17 October 2017 (UTC)


 * No, they shouldn't. Out of context search results are fairly useless.  Drop it.  --Deacon Vorbis (talk) 19:13, 17 October 2017 (UTC)


 * @Deacon Vorbis. Did you see hits in a not appropriate context? I didn't found examples of that kind (until now). So: not 'fairly useless' in this case. -- Hesselp (talk) 20:00, 17 October 2017 (UTC)


 * In order to be exact, you'd better correct the data for Books. As far as I understand, these numbers (>100...) are not really the number of results, but... I do not know what. Once again: when I've counted the results "by hand", I got much smaller numbers (23 instead of 141...). The conclusion is probably the same, but the argument more solid. Boris Tsirelson (talk) 20:17, 17 October 2017 (UTC)


 * Agree. Drop it.   Sławomir Biały  (talk) 20:51, 17 October 2017 (UTC)


 * 'Counting by hand', as proposed by Tsirel, results in the following numbers of different titles: - Google Books: "Cesaro summable sequence(s)" 43,  "Cesaro summable series" 24 - Google Scholar (articles):  "Cesaro summable sequence(s)" 150,   "Cesaro summable series" 47 Conclusion: the article should mention that "Cesàro summable sequence" and "Cesàro summable series" are used in practice. With exactly the same definition (or with equivalent definitions), and therefore with the same meaning.  -- Hesselp (talk) 22:49, 17 October 2017 (UTC)


 * I disagree with this "conclusion".  Sławomir Biały  (talk) 22:56, 17 October 2017 (UTC)
 * Why not? Given that "Zygmund's Trigonometric series uses both"... Boris Tsirelson (talk) 05:29, 18 October 2017 (UTC)
 * I don't think it is a significant aspect of the subject to mention in a relatively short article on the subject. It is certainly not as significant as one editor here is making it.  Zygmund simply uses both; he doesn't say "sometimes we also call this the sum of a sequence".  I'm open to a discussion about whether the article should mention both, but I disagree that the matter is settled simply by counting Google hits.   Sławomir Biały  (talk) 10:43, 18 October 2017 (UTC)
 * Do as Maxwell's demon: accept the acceptable, reject the unacceptable. Boris Tsirelson (talk) 11:50, 18 October 2017 (UTC)

For concluding this discussion, I suggest to change (in section "Definition")
 * The series $∑ ∞ n = 1 a_{n}$ is called Cesàro summable, ...

into
 * The sequence $\{a_n\}$ and the series $∑ ∞ n = 1 a_{n}$ are called Cesàro summable, ...

D.Lazard (talk) 12:53, 18 October 2017 (UTC)


 * I welcome Lazard's suggestion, provided that the first sentence in the section is shortened to the neutral:  Let ${a_{n}}$ be a sequence, and let  $$a_1 + \cdots + a_k = s_k$$  be its $k$th partial sum. (Or splitted into 3 lines.)


 * Although his suggestion can be improved. For the definition remains partly verbal and partly symbolic (how many readers know how to pronounce: "the series  $$\sum_{n=1}^\infty a_n$$"; could it be "the series with terms a index n" ?).   I see it as desirable to separate the verbal wording and the symbolic presentation. The definition has to tell (in words) what's the condition for a sequence/series to be called Cesàro summable. This can be accomplished (with the definition of 'Cesàro means of a sequence' restored in 'Cesàro mean' or incorporated here in 'Cesàro summation') by:   "A sequence or series is called Cesàro summable, if the Cesàro means of its sequence of partial sums tends to a limit   (the Cesàro sum of the sequence or series)."


 * An alternative without using 'Cesàro means' is (with two non-essential symbolic 'parentheses'):  "A sequence/series (with terms $$a_i$$ and partial sums $$a_1+\cdots+a_i = s_i$$)  is called Cesàro summable, if the arithmetic   mean of its first $$n$$ partial sums, $$(s_1+\cdots+s_n)/n$$,  for increasing $$n$$ tends to a limit  (the Cesàro sum of the   sequence/series)." -- Hesselp (talk) 16:31, 18 October 2017 (UTC)


 * I support Lazard's suggestion "as is", with no "provided that..." The arguments given by Hesselp and Sławomir Biały before the Lazard's suggestion do not support these additions. Accept the acceptable, reject the unacceptable. (Now I understand why Sławomir is reluctant to accept anything from Hesselp! A disillusion.) Boris Tsirelson (talk) 17:21, 18 October 2017 (UTC)
 * I just got bold and implemented Lazard's closing suggestion. Boris Tsirelson (talk) 17:48, 18 October 2017 (UTC)


 * @Tsirel. Please explain why the partial sum sk, introduced in sentence 1, should be extra specified as "partial sum of the series Σ..." but not as "partial sum of the sequence {..}" ? I cannot. Who else? The actual first sentence is as strange as:   Let $$\sum_{n=1}^\infty a_n$$ be a series, and let  $$s_k = a_1 + \cdots + a_k= \sum_{n=1}^k a_n.$$  be the $k$th partial sum of the sequence  ${a_{n}}$.  -- Hesselp (talk) 18:31, 18 October 2017 (UTC)


 * (edit conflict) No comments. (Why? Since personal attacks are disallowed. The next time I will be reluctant to support your suggestion even when it is really good, in order to avoid a new burst of your old pestered anti-series propaganda in addition to the good suggestion). Boris Tsirelson (talk) 19:13, 18 October 2017 (UTC)

I have attempted to make some copyedits to harmonize the usage in the article, and avoid confusion. Sławomir Biały (talk) 19:05, 18 October 2017 (UTC)

Remarks about "Folge" and "Reihe" in German
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)

Clueless apologies
@Deacon Vorbis, I was just in the course of repairing the mset, which I missed (apologies for this!) during an edit conflict, when I saw your edit summary, accusing me of "Full of problems: grammar, inconsistent notation with res(e)t of article, etc." BTW, the introduction of the new free variable i was intentional, to resolve existing possibilities of confusion.

I certainly do want to know where in detail you see this plethora of deficiencies in my edits. But in case that this is too much for you, I am also satisfied, if you repair (twice) the absolute rubbish of equaling an n-th element of a sequence both with its construction as with the whole sequence. I do not know what caused your harsh rebuffing. Purgy (talk) 19:58, 15 February 2018 (UTC)


 * First of all, I wasn't accusing you of anything. The edit you made introduced several grammatical errors and left the article inconsistent with respect to how sequences are denoted.  I agree the current state isn't ideal, but leaving a sequence enclosed in braces is problematic also; it can be confused with set notation.  I'm happy to convert it all to something more standard, but I'll have to get to it a little later.  In the mean time, it would at least be better to leave things consistent.  –Deacon Vorbis (carbon &bull; videos) 20:17, 15 February 2018 (UTC)


 * OK, I do not expect anymore that you substantiate to any rational level your non-accusatory remarks of my edits being "full of problems". I won't even pursue any investigations on what the "several grammatical errors" could be (as a non-native speaker I am usually highly interested in this). In any case I deny having introduced the braces-notation and any other additional inconsistencies in this article, which have not been there before. To the contrary, I removed two blatant, categorical flaws in equations (the main motivation for my edits), which, thankfully, you seem to have repaired yourself (on my hints?), meanwhile.


 * For the records, I reacted to your harsh reverting of my "good faith" (isn't this patronizing?) improvements, which were not in your style and to your taste, with a calm request for concretizing my faults, and declared myself to be satisfied by eliminating the flaws, which you re-introduced in reverting my edits to your version, but now I feel somewhat offended by your edit summary "easy, chief". Did you address yourself? Purgy (talk) 08:44, 16 February 2018 (UTC)