Talk:Cesàro mean

Vague and Potentially Confusing uses of Terminology

I find this ambiguous and confusing. I believe the problem partly relates to the established terminological difference between convergence of an infinite series and convergence of an infinite sequence, the former being no more than an instance of the latter using as "the sequence" the partial sums of "the series". This can work OK, but then, in the current wording of this article, the sentence

"If the sequence of the Cesàro means is convergent, the series is said to be Cesàro summable."

(my italics) seems to me very unclear. Which is "the series"?

Coming fresh to this terminology, and reading this sentence and what precedes it, one would be entitled to conclude that the infinite series

\sum _{i=1}^{\infty }a_{i}

is called "Cesàro summable" when the sequence {c_n}, as defined a little above, converges in the usual sense of an infinite sequence. However, referring to the page on Cesàro summation, it seems instead that it is rather a series

\sum _{i=1}^{\infty }(a_{i}-a_{i-1})

(whose partial sums -- rather than whose terms -- are the sequence {a_n}) that would properly be described as "Cesàro summable" in this case that {c_n} converges as an infinite sequence.

Can someone who is sure of established terminology -- regarding sequence/series and Cesàro summation/convergence-of-Cesàro means etc -- help here? (Or if the established terminology either does not exist or exists but is inherently confusing, then instead provide a note to warn us of this and help guide us through the minefield?)

Thanks in advance for any thoughts. — Preceding unsigned comment added by 83.217.170.175 (talk) 17:32, 17 April 2013 (UTC)[reply]

I've tried to clean cesaro-mean and cesaro-sum in the expected way. (I hope I didn't introduce more mess as has been before ...) --Gotti 05:49, 16 May 2013 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

To 83.217.170.175. After four years, I found your remark concerning the (imo: at least) unclear way 'Cesàro summability of a series' is described in Cesàro mean. I agree completely with you.
The revisions by Druseltal2005 (17 ~~April~~ May 2013) are insufficient; and he introduces the - never seen - label 'Cesàro mean of a sequence' (for: 'the limit of the Cesàro means of a sequence' or 'the Cesàro limit of a sequence').
See my recent comment and proposals on Talk:Cesàro summation. Opinions? -- Hesselp (talk) 09:09, 2 July 2017 (UTC)[reply]

The book of Konrad Knopp about infinite series is online in german and I think even in english (the latter at least in snippets at google-books). Perhaps someone can use his/this authoritative definitions from there (I've not much time & energy in the moment) Gotti 04:34, 4 July 2017 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

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