Hesselp

April 2017

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There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. Sławomir Biały (talk) 11:58, 6 May 2017 (UTC)

Oops... you'd better not violate 3RR. Boris Tsirelson (talk) 20:20, 12 May 2017 (UTC)

Topic ban

Hi. I'm just informing you that following the ANI discussion, the consensus by the community is that you be Topic baned for a period of six (6) months starting today, from editing the article at Series (mathematics) and its talk page. Any infringement of this ban will result in your account being blocked without warning for a duration of the administrator's discretion. This sanction is imposed in my capacity as an uninvolved administrator. You may appeal this sanction at the administrators' noticeboard. You may also appeal directly to me (on my talk page), before or instead of appealing to the noticeboard. The ban remains in force however, until any official change is pronounced. Kudpung กุดผึ้ง (talk) 10:35, 28 May 2017 (UTC)

Point of view

Hi, Hessel Pot. Mathematically, I like the point of view pushed by you. I happen to teach analysis only in the second year. But if I taught it in the first year, maybe I would follow you, saying "summable sequence" and "series representation", never just "series".

However, note a difference: on my courses I am the decision maker; here on Wikipedia I am not. Here a point of view cannot be presented until/unless it is widely used. And if it is, it must be presented with "due weight". Boris Tsirelson (talk) 18:52, 28 May 2017 (UTC)

Hi, Boris Tsirelson.   After a five weeks leave, I returned to a situation in which I expect to be able to communicate on internet again. As you can imagine I’m content with your support above (posted 28 May). Not with the fact that a half-year-ban can be asked for and imposed on me during my leave (announced by me in advance). But that is Wikipedia, I’ll see what to do.
Now I'm going to tell that I changed my mind on the question of how to describe the way in which the word 'series' is used in calculus.  The text I tried to import earlier (Series (mathematics), 6 May) had: "series, short for series expression"  and  "valid expression versus void expression".  But imo the word 'expression' strongly suggests that its physical symbols on paper or on screen (or physical words or sounds in case of a verbal expression) are communicating a certain idea, a notion, a value. When 'expression' is used for a group (a succession, a string) of written symbols, a reader expects that this group of symbols denotes/expresses something.  So now I think the existing practice can be described better by:
    The word 'series' is used in calculus for a combination of (symbolic, written) expressions for an infinite sequence and for     the sum function.  In case of a summable sequence such a combination is called a 'series expression', expressing
    (denoting, representing) a number or a function.
This is conform the wiki-article Mathematical notation, with:
"The notation uses symbols or symbolic expressions which are intended to have a precise semantic meaning."  and
"A mathematical expression is a sequence of symbols which can be evaluated.".
It differs from what you wrote here, 19:15, 12 May: "in general an expression has no value (but in "good" cases it has)".
-- Hesselp (talk) 10:38, 25 June 2017 (UTC)

Well, no matter what is written on "expression" on WP, I do not like this word at all. That is: in metamathematics - OK, here "expression" is on its place. But not in mathematics. Since, first of all, only countably many real numbers can be "written on paper" at all! No matter in which formal language (even if not only series, but all definitions formalizable in ZFC are allowed). In other words, a "typical" real number has no name (that is, no "writable name"). But surely, this is far from analysis. In analysis we use freely all real numbers, not only those having "personal" names of any kind (that is, "personal" definitions).

This is why I definitely dislike any use of "expression" in analysis. But, well, this is just my point-of-view; surely I cannot push it here on Wikipedia. We have a lot of articles on mathematical notions intended mostly for NONmathematicians. These could not understood and accept my point of view expressed above. Such is the life. Boris Tsirelson (talk) 15:38, 25 June 2017 (UTC)

To Boris Tsirelson.   I agree completely with your (paraphrased): "almost all real numbers don't have a writable name" . Very often I've seen that in texts on mathematics the opposite is suggested, Concerning the subject called "series", it is suggested that clever mathematicians one day will have managed to find a 'closed expression' for the sum of every infinite summable sequence.
Since some days I've on my table Calculus (ed.'91) by Swokowsky (added as reference to the Series-article on 15 April). He writes (p.534): "For most series it is very difficult to find a formula for S_n.", instead of saying:  "Forget it !".
It seems to me, that the (incorrect) intuitive feeling that all additions (even infinite ones) must/should have a 'sum', comparable with the sum of a finite number of summands at school, is the main source of the problems in reaching a description/definition of (the metamathematical/notational term) "series" acceptable for all.
How to solve this? Or let it remain unsolved? -- Hesselp (talk) 20:10, 25 June 2017 (UTC)

Nice to see that our approaches are quite close. However, the phrase of Swokowsky is about formula for partial sums, not formula for their limit. This is not related to "almost all real numbers don't have a writable name". And, I guess, he just means that an "elementary" formula for n-th term generally does not lead to an "elementary" formula for n-th sum. Of course, this claim is vague unless "elementary" is defined.

"all additions (even infinite ones) must/should have a 'sum'"? Hmmm... Does it mean "even when the terms do not tend to zero"? If so, then it is just incorrect feeling, to be corrected. Or maybe you mean a finer problem, something like this (translated to the sequence of partial sums): "every bounded increasing sequence must/should have a limit"? Surely it has (as we know); but if someone hopes to write out the limit explicitly, giving it "a writable name", then one should ask, whether the given sequence has a (writable) name. Again, this claim is vague unless "name" is defined. I can define it so that the answer is "yes", the limit has name. And the set of all "named" sequences is countable, as well as the set of "named" numbers.

Please clarify the problem (meant in "How to solve this? Or let it remain unsolved?"); for now I have (at least) two guesses, which problem is meant. Boris Tsirelson (talk) 21:20, 25 June 2017 (UTC)

To Boris Tsirelson.
I'm very sorry: I have to admit I made two blunders in my lines of 20:10, 25 June. Firstly, I intended to write "that all additions (....even infinite summable ones) must/should have a 'calculatable sum', ....".   Secondly, I've had a serious blind spot in my eye, reading Swokowsky's  "a formula for S_n"  as  "a formula for S" .
On your "Please clarify...." I can say that I thought on eventual problems in connection with attempts to replace the present non-informative  "An infinite series or simply a series is an infinite sum" (without an explanation of  'infinite sum'), by - for instance, after proposing it on Talk-page - "A combination of expressions for an infinite sequence and for the summation function, is called a series".

Two more remarks, on Swokowsky's text p.533 (ed.'91):
I.   Sw.: "a series is an expression  that represents an infinite sum",  versus
WP (referring Sw.): "a series is an infinite sum  represented by an infinite expression".
II.   Sw.: "Sometimes there is confusion between the concept of a series and that of a sequence.".   Imo it's desirable to mention in the WP-article that it is not unlikely that this confusion has to do with the following. From before 1800 until after 1950 many authors used 'series' ('Reihe', 'série', (in Russia?)) in the sequence sense. (Cauchy, 1821: On appelle série une suite indéfinie de quantités (nombres réels).)  And 'convergent' could be used for the clustering of its terms and its partial sums. (Gauss:  Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz ihrer Summirung .... . The convergence of the series itself has to be distinguished from the convergence of its summation.) Reference: S. Schwartzmann The Words of Mathematics, MAA 1994: In older usage, series sometimes meant what we would now call a sequence; for example, the Fibonacci 'series' is actually a sequence. -- Hesselp (talk) 17:05, 26 June 2017 (UTC)

About the history, I am not so interested; I let the past to remain in the past (maybe this is my drawback).

About the present. So, you say "the (incorrect) intuitive feeling that all additions (even infinite summable ones) must/should have a 'sum', comparable with the sum of a finite number of summands at school, is the main source of the problems". Now I miss your point. I feel that "infinite summable one" has indeed a sum (just the limit of its partial sums). Is my feeling incorrect? Is it a source of problems? Why really? Boris Tsirelson (talk) 19:11, 26 June 2017 (UTC)

Oops, I forgot your word "calculatable" (sum); probably this is the point that was missed by me. But please clarify, what do you mean by this word here. Boris Tsirelson (talk) 19:15, 26 June 2017 (UTC)

Probably you do not mean Computable number; under some reasonable assumptions, for a computable summable sequence, its sum is a computable number. Boris Tsirelson (talk) 19:23, 26 June 2017 (UTC)

By the way, the word "calculatable" is new for me; usually I see "calculable". Asking Internet, I got a ridiculous answer: "As adjectives the difference between calculable and calculatable is that calculable is able to be calculated while calculatable is able to be calculated; calculable." Wow! Boris Tsirelson (talk) 19:30, 26 June 2017 (UTC)

(Unindent) To Boris Tsirelson.   About the past and the present:
On the Dutch WP the dominant opinion is, that by 'the harmonic series' another mathematical concept is meant than by 'the harmonic sequence'; that the two names are not interchangeable.   Why?  Because the way the are notated is different (terms separated by comma's vs. pluses).   My contra-argument that - at least up to my school-years - the word 'sequence' ('rij' in Dutch) was almost non-existant, is rejected as not relevant now. (About Fibonacci sequence versus Fibonacci series we didn't discuss.)

I added 'calculatable' (a spontaneous invention, as a variant on your 'writable') after I saw Swokowski's sentence (p. 533):  "To calculate S₅, S₆, S₇, and so on, we add more terms of the series.".   With the word 'calculate' he suggest an action/activity (comparable with the 'addition-action' in primary school), which is not possible to perform in the general case. Imo Sw. leads the reader into a wrong direction with this 'calculate'.
The same wrong direction is suggested (imo) by the use of 'infinite sum' in the key-sentence of the series-article. The existence of a mathematical concept named 'infinite sum' is suggested, being the result of some kind of calculating/working with the given terms.
I did not at all mean 'countable number'. -- Hesselp (talk) 23:18, 26 June 2017 (UTC)

It seems I start to understand your point. Now I express my understanding, and then you'll tell me, how close it is to your idea(s).

After 25 years of teaching on math dept of Tel Aviv university I rather forgot my 40-years-old experience of teaching to engineering students in Arkhangelsk Timber institute. But now I recall it. Some students, rather innumerate, irritated me with phrases like "solve the integral", "solve the series", and even "solve the matrix". They had a misconception (or rather, a cluster of misconceptions) that a mathematical object is itself a challenge to perform some actions and get an answer.

In this sense, series are just a tip of the iceberg. Yes, given something like "12+41+5", a student performs a finite sequence of operations, gets a finite string of decimal digits and says: here is the answer. In contrast, given something like $\sum _{n=1}^{\infty }{\frac {1}{(n!)^{3}}},$ one does not perform a finite sequence of operations getting a finite string of symbols that is the answer. The same applies to something like $\int _{1}^{2}x^{\cos x}2^{\sin x}\,dx,$ or like $\cos(x\exp x)=0$ etc etc. Boris Tsirelson (talk) 05:44, 27 June 2017 (UTC)

Answering Boris Tsirelson: Your summary (05:44, 27 June) comes very close to my own ideas, yes. Just two notes.
A. In your lines  "...that a mathematical object is ..."  is followed by  "...In this sense, series are just ..." .   Can I conclude from this, that you see a series as a mathematical object ?
I agree, if you use 'series' in (one of?) its historical meaning(s):  an infinite sequence with 'additionable' terms.  But I don't agree if you use 'series' in its (more recent?) meaning:  an expression denoting the summation function for summable sequences, combined with a sequence-expression.   Such a operator-operand-couple I can see as a metamathematical object, not as a mathematical object.

B. Someone who asks "solve the series" seems to ignore the fact that a kind of 'solving' is only possible in special cases (the ones the student meets in his exercises).
In my view, that 'seems to ignore' is also applicable to the  "To calculate S₅, S₆, S₇,..."  by Swokowski.   And as well to the texts in which it is (wrongly!?) suggested that by  'an infinite sum'  and by  'a series'  a certain mathematical object is meant. (An object having: terms, partial sums and (sometimes) a sum. And sometimes being: convergent, harmonical, alternating, geometric, absolutely convergent, Cesaro convergent, monotone, telescoping, etc. etc.)
Did this kind of suggestions irritate you as well? -- Hesselp (talk) 11:15, 27 June 2017 (UTC)

A. Yes, I definitely see a series as a mathematical object. And yes, I wrote already that I dislike any use of metamatematical notions in analysis (and, more generally, wherever they are not quite relevant and can be avoided easily).

Which definition? See Equivalent definitions of mathematical structures#Mathematical practice: "A person acquainted with topological spaces knows basic relations between neighborhoods, convergence, continuity, boundary, closure, interior, open sets, closed sets, and does not need to know that some of these notions are "primary", stipulated in the definition of a topological space, while others are "secondary", characterized in terms of "primary" notions." Likewise, I know basic relations between terms and partial sums (of a series), and I do not mind, whether only terms are "primary", or only partial sums, or both. For me these are equivalent definitions. No problem.

Also, when I read (or write) the word "series" in a mathematical text, the context always makes it clear, is it a series of real numbers, or of vectors in a Banach space, or whatever. No problem, still.

B. I agree that this "kind of 'solving' is only possible in special cases". But I do not understand, why be irritated with "a certain mathematical object (having: terms, partial sums and (sometimes) a sum. And sometimes being: convergent, harmonical, alternating, geometric, absolutely convergent, Cesaro convergent, monotone, telescoping, etc. etc.)" According to my two "no problem" above, I see no problem here as well. Why do you? Boris Tsirelson (talk) 15:45, 27 June 2017 (UTC)

To Boris Tsirelson
After your last-but-one edit, I thought we were quite close. But now the sky seems to be very cloudy again. I'll try to explain why I see it like that.   You write:  "I definitely see a series as a mathematical object. ............Which definition?  See ..........For me these are equivalent definitions.".
- Please tell me: to which location in your text (or the text behind the link) your "these" refers?  I cannot find any attempt to formulate a definition of what you see as the mathematical object named 'series'. (Let alone 'definitions' in plural.)
- And please, can you write down the sentence(s) that according to you are appropriate in the WP-article to inform a student about what is meant in his calculus-lessons by 'number-series'. Supposing this reader knows already the meaning of  '(infinite) number-sequence', 'partial sums of a sequence', 'limit of a sequence' and 'summable sequence'.   I'm almost sure that in your lessons you don't say:  "a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity"  nor  "an infinite sum, represented by an infinite expression of the form .... or the form ..." (as in the present WP-article).
- Thirdly, which arguments you see against a choice for 'an operator-operand-couple' (in extenso: an expression denoting the summation-function for sequences, combined with an expression for a sequence.)   This belongs to meta-mathematics, yes; just as 'sum', 'product', 'quotient', 'logarithm', 'integral', 'root', 'power', et cetera; all having their own article in WP - for good reasons.
(I'll react soon on your analogy-edit 17:25, 27 June) -- Hesselp (talk) 20:38, 27 June 2017 (UTC)

In addition, let me note an analogy.

• Several types of numbers are in use (natural, integral, rational, real, complex); each one has its own definition; but just "number" is not used as a mathematical notion and has no definition.
• Many types of space are in use (for instance, vector space, topological space, Hilbert space, probability space); each one has its own definition; but just "space" is not used as a mathematical notion (outside high-school geometry) and has no definition. (Quoted from: v:Space (mathematics).)
• The term "random variable" in statistics is traditionally limited to the real-valued case. In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.
  However, the definition above is valid for any measurable space of values. Thus one can consider random elements of other sets, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. (Quoted from: Random_variable#Extensions.)

For random variables the general case is defined, for numbers and spaces it is not. But anyway:

• The question "prime or composite?" applies to natural numbers but does not apply to real numbers.
• The notion "dimension" applies to vector spaces and topological spaces but does not apply to probability spaces.
• The notion "median" applies to real-valued random variables but does not apply to random trees.

Now about series. Several types of series are in use (of real numbers, of complex numbers, of matrices, of functions, of vectors in a Banach space etc). Is the general case defined? I am not sure; probably not. But anyway, a series of real numbers may be convergent, harmonic, alternating, geometric, absolutely convergent, Cesaro convergent, monotone, telescoping; a series of functions may be convergent pointwise, uniformly convergent, but not harmonic nor geometric... Is this a problem? Really, not. Every occurrence of the word "series" in a mathematical text has a well-defined meaning specified (explicitly or implicitly) by the context. Just like every occurrence of the word "number" (or "space", or "random variable") in a mathematical text. Boris Tsirelson (talk) 17:25, 27 June 2017 (UTC)

To Boris Tsirelson, on your text 17:25, 27 June:
Regarding 'space' I agree that this term occurs in a broad variety of combinations. Each combination needs (and has in WP) its own definition.
Regarding 'number', the situation is quite similar. (I should be happy with different names for different types of 'numbers' - in French you have  numéro  and  nombre , to make a start.)
Regarding 'sequence' (not named by you) the situation is different: there is a general agreement on the mathematical meaning of this word.  Notwithstanding the fact that a sequence of real numbers can have properties of another kind than a sequence of functions.

Now about 'series'.   Given the fact that - as you say - there is probably not a definition for the general case, a definition for the special case of (say) 'a series of real numbers'  is badly needed.  Please tell us: what do you say in your lessons in this case?
On your sentence:  "Every occurrence of the word "series" in a mathematical text has a well-defined meaning specified (explicitly or implicitly) by the context.".  I don't think I can agree with every occurrence, but in many cases I think I can.   Last question: Can you describe that "well-defined meaning" with other words than  "an expression denoting the summation function for sequences, combined with an expression for a sequence" ? -- Hesselp (talk) 23:18, 27 June 2017 (UTC)

Yes, for now I feel the same: that sometimes I understand you readily, sometimes not.

Here are three different questions:

• Question M: How do I interpret the word "series" when reading or writing an article in a mathematical journal?
• Question S: How do I define "series" on a lecture for students?
• Question W: What to write about series on a wiki?

For now I discuss Question M only. The answer to Question S depends heavily on the audience of students. Math dept of Tel Aviv univ is one audience; Arkhangelsk timber institute is another. The answer to Question W depends heavily on a wiki: policies of Wikipedia are very different from these on Wikiversity. Anyway, I want to first reach mutual understanding on Question M, and (if successful) turn afterwards to harder Questions S and W.

• Definition 1. A series (of real numbers) is a pair of (infinite) sequences $a_{1},a_{2},\dots$ and $S_{1},S_{2},\dots$ (of real numbers) such that $S_{n}=a_{1}+\dots +a_{n}$ for each $n$.
• Definition 1'. (a) A series (of real numbers) is a (infinite) sequence $a_{1},a_{2},\dots$ (of real numbers); (b) $S_{n}=a_{1}+\dots +a_{n}$ for each $n$.
• Definition 1''. (a) A series (of real numbers) is a (infinite) sequence $S_{1},S_{2},\dots$ (of real numbers); (b) $a_{n}=S_{n}-S_{n-1}$ for each $n$ (where $S_{0}=0$).

These are equivalent definitions. In every case:

• Definition 2. $a_{n}$ are called terms, and $S_{n}$ are called partial sums.
• Definition 3. A series is called convergent, if partial sums converge to a (finite) limit. This limit is called the sum of the (convergent) series.

And so on... Boris Tsirelson (talk) 05:09, 28 June 2017 (UTC)

To Boris Tsirelson. For me, my central point is question W:  What should be written at the moment the word 'series' appears in the article with Series as heading;  in Wikipedia (not 'a wiki');  focused on the non-generalized case of a 'real-number-series';  focused on calculus, not on algebra;  supposing the reader is familiar with '(infinite) number-sequence', 'partial sums of a sequence', 'limit of a sequence' and 'summable sequence'.   But this preference doesn't prevent me to state the following points.

Points I and II

- I.  Are you aware of the fact (is it a fact?) that the word 'series' is very often used in direct connection with (mostly directly followed by) a symbolic, written form of type 'capital-sigma' or type 'pluses-bullets'
( Σ_n=1^∞$a_{n}$  or  $a_{1}+a_{2}+a_{3}+\cdots$  or variants).
- II.  As a consequence of I, the word 'series' will be absent in the oral/verbal part of your lectures on calculus (without a blackboard or a more modern pictural divice for communication).  I don't have a verbal alternative for a capital-sigma-form or a pluses-bullets-form. -- Hesselp (talk) 21:43, 28 June 2017 (UTC)

Now I feel like a person that came to discuss a matter of number theory, say "are there infinitely many prime numbers?", but is suddenly asked, "are you aware of the fact that the word 'integer' is very often used in direct connection with arabic numerals or roman numerals?". It seems we agreed to put metamathematical notions aside (wherever possible). For me, there is no "written form of type 'capital-sigma' or type 'pluses-bullets'" in analysis. Just as there is no "written form of arabic or roman type" in number theory.

"As a consequence of I, the word 'series' will be absent in the oral/verbal part of your lectures on calculus (without a blackboard or a more modern pictural device for communication). I don't have a verbal alternative for a capital-sigma-form or a pluses-bullets-form." Sorry, I miss this point completely. First, I cannot imagine myself lecturing without a blackboard. But anyway, surely I could pronounce the definitions (that I wrote above) on a lecture, thus using the word "series". And after that I could teach the usual theory of series. Boris Tsirelson (talk) 05:23, 29 June 2017 (UTC)

Well, of course, at some point I would say (on the lecture): We denote by $a_{1}+a_{2}+\dots$ or $\sum _{n=1}^{\infty }a_{n}$ the series whose terms are $a_{1},a_{2},\dots ;$ and its partial sums are $S_{n}=\sum _{k=1}^{n}a_{k}.$ Boris Tsirelson (talk) 06:09, 29 June 2017 (UTC)

An important reservation that I should say in the next minute: However, the sum of a (convergent) series is also denoted by $a_{1}+a_{2}+\dots$ or $\sum _{n=1}^{\infty }a_{n};$ this is of course an ambiguity; the context help us to understand (in each case separately), does the author means the sum of the series or the series itself. Boris Tsirelson (talk) 18:48, 29 June 2017 (UTC)

My 'Point I' concerns written text, my 'Point II' oral/spoken text.  I observe that the role of the word 'series' is different in this two situations.  Please, try to react on this aspect - see my 'Point X' below.  How do you SAY: $\sum _{n=1}^{\infty }a_{n}$  ?
-- Hesselp (talk) 23:07, 29 June 2017 (UTC)

Points III and further

- III. Can you imagine that I see the facts described in I and II as a strong indication that the word 'series' is not in use for a mathematical concept, but for a notational form?

- IV.  With your three definition-variants (1, 1', and 1") you want to illustrate that, in your view/interpretation, the terms-sequence (a_n) and the sums-sequence (S_n) have an equivalent role in the concept/object 'series'.  Right?     I paraphrase your triple by:
"Definition: An ordered pair of real-number-sequences a_n; b_n ,  related by  b_n = a₁ + ··· + a_n  (or the equivalent:  a₁ = b₁,   a_n+1 = b_{n +1} − b_n ),  is called a (real-number-)series."   Right?
- V.  Why this emphasis on the equivalent role of both sequences?   The relation between a (summable) terms-sequence and the limit of its sums-sequence is an important one in calculus. As is the relation between a (complicated) function and the sequence of its (less complicated) Maclaurin-terms. As well as the relation between f and the sequence of the Fourier-terms of f. On the contrary, the relation between the sums-sequence and the limit of its terms-sequence has no applications in calculus (as far as I know).  So I don't see a good reason to emphasize an equivalency of terms-sequence and sums-sequence.  Right?
- VI.  I've another argument against adopting the pair of sequences as being the heart of a 'series'-concept.  Isn't it standard use in mathematics to formulate definitions as simple/elementary as possible? So why should the sums-sequence be mentioned in the definition, while this sums-sequence is already completely determined by the other sequence?
- VII.  Your pair-of-sequences as being the mathematical meaning of 'series', I met earlier in calculus-books by: Creighton Buck 1956, Zamansky '58, Apostol '74, Maurin '76, Protter/Morrey '77, Encyclopaedia of Mathematics '92, Gaughan '98, Boos '00, Edward Azoff '05. Not long ago I red that the source is in a Bourbaki-publication. (I cannot find back where I saw this; do you know this source?)  As I said in VI, this definition doesn't say anything more than 'series' is another word for 'sequence'. Or more precise: A sequence with a sums-sequence (the same as: a sequence with additionable terms) is called series.
Or can you describe a difference between "a terms-sequence  sums-sequence  pair"  and  "a sequence with terms that can be added to form its sums-sequence" (essentially the same as Cauchy's: "une suite de nombres réels") ? -- Hesselp (talk) 21:43, 28 June 2017 (UTC)

- VIII.  Once more on your:  "How do I interpret the word "series" when reading or writing an article in a mathematical journal?".
Can you find back places in articles/books written by you (or red by you) where you used/red the word 'series' ? (The articles/books not being tutorial texts on calculus.)   Can at that places the word 'series' be red as: "the combination of expressions for the summation function and for a sequence"?  Or can "series Σ a "  (or "series a₁ + a₂ + ··· ")  be red as "sequence a" ? -- Hesselp (talk) 05:36, 29 June 2017 (UTC)

(I wrote my 'point VIII' before your '(Unindent)' came in, but was just some minutes late in posting it.)
- IX. On your:  "It seems we agreed to put metamathematical notions aside (wherever possible).":  I see this put aside as not possible in a discussion on the way mathematicians use the word 'series' in a calculus context.   The consequence of this put aside should be (imo) that you want to skip WP-articles with titles: 'sum', 'product', 'quotient', 'logarithm', 'integral', 'root', 'power', et cetera.
- X.  On your:  "...surely I could pronounce..."  and your  "at some point I would say [!,HP]...":  Do you really say (verbally, in words), with your back to the blackboard:  "series Greek capital letter sigma indexed by n is one and by infinite a indexed n " ?  Or something like this, starting with the word 'series' ? -- Hesselp (talk) 07:03, 29 June 2017 (UTC)

Sorry, it is impractical to discuss 8 points in parallel. Let us close one point and then start another. Boris Tsirelson (talk) 06:27, 29 June 2017 (UTC)
Yes, it can be practical to discuss points I - X (maybe some combined) in different (sub)sections. -- Hesselp (talk) 07:03, 29 June 2017 (UTC)
Well, then create the subsections; but anyway I do not want to scatter, and will attend one subsection at a time. The more so that the conclusion reached in one subsection may affect the discussion of another. Thus, choose their order... Boris Tsirelson (talk) 07:11, 29 June 2017 (UTC)</math>
I leave it to you, Tsirel, to choose the order.   Or else: why not the order of the roman numbers? -- Hesselp (talk) 08:09, 29 June 2017 (UTC)
OK, I start the split process. Boris Tsirelson (talk) 09:14, 29 June 2017 (UTC)

Maybe I understand

Maybe I understand your point, at last!

Here is my guess. We have here another oddity of mathematical terminology. (Strangely I did not note it before.) First, both notations, $a_{1}+a_{2}+\dots$ and $\sum _{n=1}^{\infty }a_{n},$ are ambiguous; depending on the context they may denote either a series (that has terms and partial sums), or the sum of this series! Examples:

(a) the series $\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}}}$ converges absolutely;

(b) according to Euler, $\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}};$

in (a) the expression denotes the series, but in (b) a similar expression denotes the sum of the series.

Is the word "series" ambiguous in the same way? Probably, less ambiguous, since usually one writes "sum of the series", not just "series", when the sum is meant. But probably a less accurate language is used sometimes, like this:

(c) the series $\sum _{n=1}^{\infty }{\frac {0.9^{n}}{n^{2}}}$ does not exceed $\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}};$

in this phrase the "series" does not exceed a number, therefore the "series" is interpreted as a number. Though, one may say that no, the author uses element-wise inequality between one series and another series, and deduces inequality between their sums.

Does a similar oddity apply to integrals? Usually, $\int _{0}^{1}x\,dx$ denotes the number ${\frac {1}{2}}$ (rather than the whole integrand function). But probably a less accurate language is used sometimes, like this:

(d) all Riemann sums for $\int _{0}^{1}x\,dx$ are less than 1;

(e) the integral $\int _{0}^{\infty }{\frac {\sin x}{x}}\,dx$ is not absolutely convergent;

surely in (d) one does not mean Riemann sums for the number ${\frac {1}{2}},$ and in (e) one does not mean that a number could be absolutely convergent. Boris Tsirelson (talk) 18:59, 29 June 2017 (UTC)

Tsirel, you write (18:59, 29 June) on the (regrettable) fact that the capital-sigma form (and the pluses-bullets form) is used to denote a number as well as a not-number (a 'series', whatever it may be).  In the WP-article you can see this double-meaning, just before the subhead 'Convergent series'.  Many authors use the same forms also in a third meaning: to denote the sum sequence of a given sequence.
But this is NOT my point (not one of my points). What I want is: to make clear in the WP-article why a student should believe in the existence of a 'mathematical object' (a 'series') somewhere between:  a sequence, its sum sequence, and its (eventual) sum. Why should students have to learn to manipulate with this (difficult to define: see the wealth of contradicting sources) 'object' ? -- Hesselp (talk) 23:13, 29 June 2017 (UTC)

I see. Well, then we basically return to the beginning of this section:

Mathematically, I like the point of view pushed by you. I happen to teach analysis only in the second year. But if I taught it in the first year, maybe I would follow you, saying "summable sequence" and "series representation", never just "series".

However, note a difference: on my courses I am the decision maker; here on Wikipedia I am not. Here a point of view cannot be presented until/unless it is widely used. And if it is, it must be presented with "due weight".

Locally, on many points I fail to understand you. Nevertheless, globally I (seem to) understand, and to some extent I tend to agree that the language of the series theory could be better.

Now the question is, how can you push your point of view. Here on Wikipedia the argument that "your" language is better does not work. Wikipedia mirrors the current state of the society, not a desirable better state. This itself means that you cannot use WP. In addition, in my opinion, you often intermix correct and incorrect claims, which irritates other editors (as you definitely feel) and makes WP useless for you.

I invite you to Wikiversity (WV). Yes, it is much less visited than WP. However, it is possible to provide a link from a WP article to a relevant WV article (if the WP community does not object, of course); this option is rarely used, but here is a recent example: the WP article "Representation theory of the Lorentz group" contains (in the end of the lead) a link to WV article "Representation theory of the Lorentz group".

In fact, "08:42, 21 June 2015 User account Hesselp (discuss | contribs) was created automatically" on WV, see v:Special:Log/Hesselp. Thus, I guess, you can log in there with the same password as here. And then we could continue the discussion there, either on your talk page, or on my talk page.

Or, better, you could create an article there, intended to help students (which is welcome there much more than here). Another option is, to submit your article to WikiJournal of Science there. (I once did so, see it.) Boris Tsirelson (talk) 05:46, 30 June 2017 (UTC)

@Tsirel   Hi Boris.   After three monthes finally a reaction on your constructive words above.
Your suggestions concerning Wikiversity and WikiJournal of Science I certainly keep in mind. But at first I changed over to the German WP. You can read that language? If so, perhaps you can have a look in de:Reihe (Mathematik), de:Folge (Mathematik), de:Differenzenfolge and de:Cesàro-Mittel (plus Diskussion pages). There I found very clear contradictions in the existing texts.
You wrote:  "WP mirrors the current state of the [mathematical] society ....". Okay. But in that 'current state' the word 'series' (and 'Reihe' and 'série') is used (and has been used) in a number of different meanings. So a WP-article should (imo) try to describe that situation.
One more point. You supported (?) the use of 'summable sequence' instead of the more usual "sequence with a convergent associated series" (or some kind of abbreviation of this).   Unfortunately I met some places where 'summable sequence' means (if I got it right; in contexts with 'indexed families'): "sequence with converging partial sums of its absolute (unsigned) terms".   We have to be carefull in using this word 'summable'.   Hessel -- Hesselp (talk) 21:12, 26 September 2017 (UTC)

Ironically, I learnt German during 15 years (pupil, univ. student, doctorant) and never used; I never formally learnt English, but always used; and now my Deutsch is much worse than my English. Well, I'have read (not thoroughly) de:Reihe (Mathematik) (especially "Semantik und Vergleich"), and I see that the situation there is quite similar to the situation here. No wonder; both mirror the same state of the society. About "WP-article should try to describe that situation" I agree; but it is not about contradictions in mathematics (God forbid...), but about oddities of mathematical terminology. A mathematician that reads or writes a text that uses "series" always guesses correctly from the context what is meant this time, and is never confused. You wrote above: I don't think I can agree with every occurrence, but in many cases I think I can. No, I really cannot imagine myself, or my colleague, to be seriously confused this way. A student may be confused, yes, this is a pedagogical problem, not a mathematical problem. Every natural language has oddities; native speakers do not suffer from this, foreigners do. Mathematicians use in practice a "half-natural" language, which is rather safe, since there is a formal language behind, with no oddities, but too unpractical for everyday use.

Yes, I know that "summable family" (of real numbers) is automatically "absolute", just because no order is given on the index set. But the word "sequence" (in contrast to "family") emphasizes the given order of terms. Thus I would not hesitate (on my course) to say: each sequence is a family, but be careful: a summable sequence need not be a summable family. Boris Tsirelson (talk) 05:37, 27 September 2017 (UTC)

Only a pedagogical problem?

@Tsirel
A.   "it is not about contradictions in mathematics [...] but about oddities of mathematical terminology"
A mathematical or a pedagogical problem? I cannot choose. But in the present text of de:Reihe (Mathematik) I read in Satz 3 that 'Reihe' is the name for every Folge with a Differenzenfolge (e.g. every Zahlenfolge). And in Satz 8 (in a cripple way) that 'Reihe zum Folge (a_n)' means the same as 'Partialsummenfolge der Folge (a_n)' . Sätz 3 and 8 are contradictional, imo. At least it should be noticed that this are two different ways the word 'Reihe' is used in calculus.

B.   How do first class mathematicians (try to) solve this definition-question? See:
- Bourbaki, Éléments de Mathématique, Première partie, Livre III (Topologie générale), Chap. 3, Par. 4, N^o 6. Series (Deuxième Édition, 1951, p. 42-43)  "On appelle série définie par la suite (x_n) le couple des suites (x_n) et (s_n) ainsi associées."
- Encyclopedia of Mathematics - Series:   "A pair of sequences of complex numbers {a_n} and {a₁+ ··· + a_n} is called a (simple) series of numbers"
- B. Tsirelson 28 June 2017:   "Definition 1. A series (of real numbers) is a pair of (infinite) sequences $a_{1},a_{2},\dots$ and $S_{1},S_{2},\dots$ (of real numbers) such that $S_{n}=a_{1}+\dots +a_{n}$ for each $n$."

This definition - though formally not incorrect - is absurd, for you can choose whatever you want as second element in the pair. "Series" can be defined as being an ordered pair with an infinite sequence-with-addition as its first element and a arbitrary object as its second. Consequences:
an alternating/harmonic/Fibonacci series $\equiv$ a pair with an alternating/harmonic/Fibonacci sequence as its first element;
a convergent series $\equiv$ a pair with as first element a sequence with converging partial sums;
the terms / partial sums of a series $\equiv$ the terms / partial sums of its first element
the sum of a series ( (a_n) ; .... )   $\equiv$ the limit of the partial sums of sequence (a_n).
A series is a pair?  Did you ever see  (1, 1/2, 1/4, 1/8, ... ;  1, 3/2, 7/4, 15/8, ...)  as an example of a convergent series?

C.   "Experienced mathematicians don't have problems with the meaning of 'series'. ".
Yes, they can distinguish between a convergent sequence and a convergent series. They know that 'sum of a sequence' means (about?) the same as 'sum of a series'. And they know that you should never say 'absolutely convergent sequence'. Nor 'limit of a series'.  But do they all know the difference between a Cesàro-convergent sequence and a Cesàro-convergent series ?   See the last sentence before the heading "Examples" in [1]: "For any convergent sequence, the corresponding series is Cesàro summable and the limit of the sequence coincides with the Cesàro sum."

D.   Is it a mathematical problem or a pedagogical problem that caused the situation that writers and controllers of the English WP-article on 'series', gave up their attempts to formulate a definition?  I don't know.

E.   On the meaning of 'summable sequence': Bourbaki choose for summable = absolutely summable !  For evidence, scroll down to the last five lines of page 269 in this book. (The same in the first (French) edition, 1942.)

F. About writing an article on this subject: ten years ago I had this article in the journal "Nieuw Archief voor wiskunde" of the Dutch Mathematical Society (title: Was Reihen sind, kann man nicht sagen; reeks = series, rij = sequence)
-- Hesselp (talk) 20:43, 29 September 2017 (UTC)

I do not like to discuss a lot of arguments at once. For me, if something is true, it is true due to a single strong argument, not a lot of weak arguments.

About your "A series is a pair?  Did you ever see  (1, 1/2, 1/4, 1/8, ... ;  1, 3/2, 7/4, 15/8, ...)  as an example of a convergent series?" I may reply: did you ever see ({1,2},{(1,1),(1,2),(2,2)}) as an example of an ordered set? Or ({1,2},{{},{1},{1,2}}) as an example of a topological space? Now try to write in this way an example of a two-element group; it will be more cumbersome. I could also use the definition (a,b)={{a},{a,b}} in order to make the things much worse looking. Not a valid argument against a definition. Or else most of our definitions will be under attack. Boris Tsirelson (talk) 05:24, 1 October 2017 (UTC)

Showing the symbolic form  (1, 1/2, 1/4, 1/8, ... ;  1, 3/2, 7/4, 15/8, ...)  denoting a (Bourbaki-)series, intended to illustrate the absurdness (not the formal wrongness) of the Bourbaki-definition. So I can agree with your comments.
G.   The artificial nature of defining 'series' as a pair, I see reflected in the number of inconsequences in the 'Encyclopedia of Mathematics'. For instance:
G1. Series,ca line 24 "An example of a convergent series is the sum of the terms of an infinite geometric progression"
A series is a pair, the sum of a progression is not a pair.
G2. Power series, ca line 21  "When r = ∞, the series (1) either terminates, i.e. it is a polynomial ...,"
A series is a pair, a polynomial is not a pair.
G3. Uniformly-convergent series, ca line 15  "Example. The series Σ zⁿ/n! = e^z "
A series is a pair, e^z is not a pair.
G4. Geometric progression, ca line 10  "The expression [...] is the simplest example of a convergent series"
A series is a pair, an expression is not a pair.
G5.  Multiple series, line 1 "s-tuple series: An expression of the form ..."   Idem.

H. All such oddities are caused by the fact that Cauchy in 1821 (French original page 123) choose for 'convergente' to label a sequence with converging/clustering partial sums. His essential difference between "having converging terms" and "convergent" (= "having converging partial sums") is (or: seems to be) overlooked many times.
Nowadays everyone can verify this: scroll to page 85, textlines 1 and 6. The Cauchy-meaning of 'quantity' is explained in page 5, textline 8-9 saying: "we will only apply the term quantities to real positive or negative quantities, that is to say to numbers preceded by the signs + or − ."
The difference Cauchy made between 'sequence' (French: suite) and 'series' (French: série) is, imo, in modern words:
sequence: a mapping on N, often in a set with a 'distance',
series: a mapping on N in a set with a 'distance' plus an 'addition'.

The Cauchy-difference between sequence and series reflects the difference in content of the chapter headed by 'Sequences' and the chapter headed by 'Series' in very many textbooks on Calculus. (In chapter 'Sequences' any addition of terms is absent.)
Do you agree with me that it is much more practical to consider the result of the Taylor-expansion of a function f (and the result of its Fourier-expansion as well) , named "the Taylor series of f", as a mapping-on-N  instead of a pair of two mappings-on-N ?

B1. A supplement to my earlier point B on the not very convincing Bourbaki-definition:
Instead of Bourbaki's 'series' as name for a pair  ( {a_n} ;  {a₁+ ··· +a_n} ), you could use the word 'series' for the quartupel
( {a_n} ;  {a₁+···+a_n} ;  {|a₁|+···+|a_n|} ;  {a₁/1+(a₁+(a₁+a₂))/2+···+(a₁+(a₁+a₂)+···+(a₁+···+a_n))/n} ).
Now with the simple/short descriptions:   a convergent / absolutely-convergent / Cesàro-convergent series  $\equiv$  a series with a convergent 2nd / 3rd / 4th element. (To avoid misunderstanding: I'm not advocating this variant.)
-- Hesselp (talk) 20:09, 2 October 2017 (UTC)

About "illustrate the absurdness (not the formal wrongness) of the Bourbaki-definition": so, do you find most of our definitions to be "absurd (but not formally wrong)"? Or only that of series? Specifically: do you find absurd the definition of (a) ordered set, (b) topological space, (c) group?

About all else: well, this discussion becomes senseless. You insist on "literal" treatment of texts about series, while I support the (widely used) context-dependent treatment. We must agree to disagree. I know that math is formal; but I also know that in practice it is rather formalizable. It is written on semi-natural language(s); a mathematician can translate it to a formal language (using his understanding of context), but this is very seldom made, since it is (a) as tedious as programming, and (b) of almost no practical use (unlike programming); really, this is made only for proof assistants. Boris Tsirelson (talk) 20:44, 2 October 2017 (UTC)

By the way, your discussion style reminds me of that of Chjoaygame (long series of arguments against the mainstream, based on old texts of founding fathers, etc). Maybe you two could be a kind of friends... Boris Tsirelson (talk) 05:59, 3 October 2017 (UTC)

About your first three lines.   My qualification 'absurd' (unnecessary complicated) had to do ONLY with the introduction of sequence - sum sequence - pairs as a notion apart from sequence and sum sequence of a sequence. And with using the word 'series' exclusively for such pairs in texts on calculus.  I cannot see that I wrote or suggested that this qualification had to do with anything else.
Unfortunately you don't comment substantially on my claim 'unnecessary complicated'.

About: "long series of arguments against the mainstream, based on old texts of founding fathers,...".   I'm puzzling on what you could have had in mind when you typed the mainstream: (a) Do you see a certain way of defining (describing of the content of) the word 'series' that is used by the main part of authors of textbooks (and WP-articles) on calculus? Which way?   (b) Or do you mean that most of these authors don't present a formally correct definition being in line with the way they use the word 'series' in their subsequent text?   (c) Or a third possibility?
Moreover, is your clause "based on old texts of founding fathers" referring to 'mainstream' or to 'arguments'? Which 'fathers' are meant?

About your 'literal' versus 'context-dependent'.  When and where I see possibilities to avoid/reduce inconsistancies and ambiguities concerning the use of the word 'series' in WP-texts, in a way that doesn't make the text more difficult to read for beginners, I want to discuss and finally import such (what I see as) improvements. In other words: making a text less 'context-dependent' without lowering the readability, I see as improvement. You agree with this? Could this be an alternative for your 'agree to disagree'? -- Hesselp (talk) 19:58, 3 October 2017 (UTC)

About "you don't comment substantially on...": if you'll single out one argument, I'll be able to concentrate on it. I am lost among ABCDEFG(-5)H...

Have you an example when the usual "series" terminology really confuses professional mathematicians (rather than students)? Without such example I still think the problem is pedagogical.

About your last paragraph: well, yes and no. As I wrote, I could use such improvement on my course, since (a) the course is a pedagogical effort, (b) on my course I am the decision maker. However, I would not do so in a research paper (see "a"), nor on Wikipedia (see "b").

About "founding fathers" I am sorry; you mention once Cauchy 1821, but you do not base on such old works; my error. Boris Tsirelson (talk) 09:41, 4 October 2017 (UTC)

Almost all the points marked A-H are just observations by me, related with the unclearness of the word 'series'; only the last sentence in H, I formulated as a direct question to you.

On the example you asked for: see the six sentences in the section 'Series' in the WP-article Sequences:
- "A series is, informally, an expression of the form Σa_n "
- "The partial sums of a series are the expressions ...."
- "...we say that the series Σa_n is convergent, ..."  [a convergent expression?? what's that?].
Will professional mathematicians be able to decipher the intended content of this section? Probably yes, but only with quite a bit of their experience as 'context'. I see at least as many 'mathematical' faults in this text than 'pedagogical' faults. [The central point in my critics is in fact: don't see notation as content.]
Now you are going to say that you didn't mean textbooks or WP-articles, but high standard research papers. Okay. Already for a long time I try to find occurrences of the word 'series' in this kind of texts, but I find it very hard to spot them. Are they rare, or do I search at the wrong places?
On the examples G1-G5 from the Encyclopedia of Mathematics: I would say that this are more mathematical inconsequencies than pedagogical weaknesses. And in what degree your professional mathematicians will be really confused by them? Degree zero?  Is improvement desirable?

'Founding Father' Cauchy once opened my eyes: page 123 sentence 1 and 4. This two sentences are the base for me (except his unfortunate choice for 'convergent' for converging partial sums). And I would say I suppose that Cauchy's formalisation is much closer to the informal notion of a series at the present public, than Bourbaki's artificial sequence-sumsequence-pairs.
Cauchy's Cours d'Analyse is an "old work", indeed. But his approach is explicitely copied until at least 1938: G. Kowalewski, Die klassische Probleme der Analysis des Unendlichen. -- Hesselp (talk) 21:25, 4 October 2017 (UTC)

Well, the last sentence in H:

Do you agree with me that it is much more practical to consider the result of the Taylor-expansion of a function f (and the result of its Fourier-expansion as well) , named "the Taylor series of f", as a mapping-on-N  instead of a pair of two mappings-on-N ?

No, I do not. And really, you already know, why; we are nearly in a loop. I quote myself (this page, far above):

See Equivalent definitions of mathematical structures#Mathematical practice: "A person acquainted with topological spaces knows basic relations between neighborhoods, convergence, continuity, boundary, closure, interior, open sets, closed sets, and does not need to know that some of these notions are "primary", stipulated in the definition of a topological space, while others are "secondary", characterized in terms of "primary" notions." Likewise, I know basic relations between terms and partial sums (of a series), and I do not mind, whether only terms are "primary", or only partial sums, or both. For me these are equivalent definitions. No problem.

You see, mathematicians deal with notions, not definitions. The point is that a notion has many equivalent definitions. I can say that a notion is an equivalence class of definitions. Definitions are forgotten after learning basics. This is why mathematicians understand each other in spite of different (but equivalent) definitions used by different lecturers or textbooks.

Ask mathematicians "do you know the notion 'series'?"; they will answer "sure". Ask them a "yes/no" question about series (notion, not definition!), for instance, "if a Fourier series converges in $L_{2}[0,2\pi ],$ does this imply convergence almost everywhere?" They all will give the same answer. (Well, in practice, some will reply "I do not know", or "sorry I am too busy", or tomorrow "oops, I was mistaken yesterday, sorry"; but ultimately, they will agree.) Surely I can reformulate this question getting rid of the notion "series", but will it be more practical or less practical?..

Identify a series with the sequence of its terms, or the sequence of its partial sums, or with the pair of sequences, it is all the same, due to the evident one-to-one correspondence between these objects. No one will bother. Likewise, try asking a number theorist (that thinks, say, about the distribution of primes), is it more practical to represent numbers by decimal digits, or binary digits, and he'll answer: do I need digits? I think about numbers, not bothering about strings of digits.

Only a pedagogical problem. Mathematicians use a single notion "series" and need not bother. Students learn different (but equivalent) formalizations of this notion, and might like one and dislike another. Boris Tsirelson (talk) 16:41, 5 October 2017 (UTC)

On your remarks (5 October 2017) until "Identify ..." :   No problem for me to agree with you that a mathematical notion can be defined in differntly formulated - equivalent - ways.
On your sentence:   "Identify a series with the sequence of its terms, or the sequence of its partial sums, or with the pair of sequences, it is all the same, due to the evident one-to-one correspondence between these objects."   I read this as:
A series can be identified/represented by its terms,
a series can be identified/represented by its partial sums, and
a series can be identified/represented by its terms-sums-couple.
But the mathematical notion mostly called 'series' isn't defined by showing its terms representation. Nor by showing - imo - its partial-sums representation or its terms-and-sums representation.
One more point: I hope I don't misread your sentence by interpreting 'these objects' as: 'these representations' or 'these identification tools'.

Substituting the word 'series' in your sentence by the word 'sequence', the content of it remains equally true - yes?  This seems to indicate that you see the notion series as identical with the notion sequence - yes?
If your answer is 'yes', I agree with 'Only a pedagogical problem' (without question mark). I.e.: the problem of how to convince quite a lot of authors of calculus-books (and WP-articles) that their way of presenting this notion-with-two-names in their separate chapters 'Sequences' and 'Series' makes it very difficult for students to see the definitions in the two chapters as intended to be equivalent. -- Hesselp (talk) 21:56, 8 October 2017 (UTC)

No. Once again, "I know basic relations between terms and partial sums (of a series), and...".   I definitely do not want to define "terms and partial sums of a sequence", "Fourier sequence" etc. I definitely do not want to rewrite the (example of) question "if a Fourier series converges in $L_{2}[0,2\pi ],$ does this imply convergence almost everywhere?" as "if a Fourier sequence converges in $L_{2}[0,2\pi ],$ does this imply convergence almost everywhere?" The existing terminology is just convenient; even if it may look a bit strange when considered at a certain angle, this is not enough motivation for abandoning it. Just another oddity of the math language (practical, formalizable, not formal). Like many persistent oddities of a natural language, it is more convenient to keep it than to abandon it. Simple as this. Boris Tsirelson (talk) 18:16, 9 October 2017 (UTC)

Your answer on my  "you see the notion series as identical with the notion sequence - yes?"  is  "No.".
With as your argument ('argument'):  "The existing terminology is just convenient;"  and (paraphrased): there is no reason to abandon "Fourier series" and change over to "Fourier sequence".
Sorry, but I cannot see this remarks as an explanation of the difference between 'the notion sequence' and 'the notion series'. For me 'the notion sequence' coincides with 'the notion mapping on N'; and for 'the notion series' you didn't show me a - mathematically different - alternative.
To avoid misunderstanding: I accept the situation that the (worldwide) tradition has 'sequence' in some contexts, and 'series' in other contexts (quite often it is 'series' when the question of convergence of the partial sums of the terms is actual). And sometimes the tradition has no preference (harmonical sequence / harmonical series).   Two names for the same notion: it's not the ideal situation, but tradition is stronger than Esperanto.

In the Encyclopedia of Mathematics I found as headings:
- Monotone sequence vs. Alternating series
- Geometric progression vs. Hypergeometric series
- Arithmetic progression vs. Arithmetic series of order m
- Cauchy sequence vs. Harmonic series
Can you explain the reason for the choice between 'sequence/progression' and 'series' in each case? (I doubt; I couldn't find the clue in the articles Series and Sequence.)
Two names for the same notion, okay, but be consequent for the rest. For a student and a WP-reader is unhappy with (cited from the same encyclopedia):
- Harmonic series:   The series of numbers Σ 1/k
- Euler series:   The expression Σ 1/p
- Arithmetic series of order m:   The sequence of values ...
- Series (= infinite sum):   The pair of a sequence its sum sequence. -- Hesselp (talk) 00:09, 10 October 2017 (UTC)

Oops, I am wrong. There is no problem to say "terms and partial sums of a sequence". However, what about "Fourier sequence"? Is it the sequence of terms of a Fourier series, or of its partial sums? Boris Tsirelson (talk) 19:38, 9 October 2017 (UTC)

You will understand that I agree with your correction. About your Fourier-question:
The result of the Fourier-expansion of a given function f is (in my perception) the mapping on N that combines each element of N with a term (a function, not a number) of the expansion. Whether you name this result "the Fourier series of f" or "the Fourier sequence of f" ist mir egal. Although I know that the second is extreme unusual, as is its symbolic expression with commas instead of pluses; but as I'm not aware of a mathematical difference in notion, the name with 'sequence' is not mathematically incorrect. -- Hesselp (talk) 00:15, 10 October 2017 (UTC)

Well, frankly, we could identify a series with the sequence of its partial sums, say "differences" whenever we now say "terms", and say "terms" whenever we now say "partial sums". Then the above (example of) question sounds "if a Fourier sequence converges in $L_{2}[0,2\pi ],$ does this imply convergence almost everywhere?"; rather acceptable. And the Alternating series test becomes: "for a sequence to converge it is sufficient that its differences are alternating and monotonically decreasing to 0 in absolute values"; rather acceptable, too. But who is motivated enough to replace a lot of habitual formulations with such new formulations? Also, I am afraid that, trying to commit such a terminological revolution, we'll divide in two parties, one identifying a series with the sequence of its partial sums, the other identifying a series with the sequence of its terms. Isn't it better to leave it as is? Boris Tsirelson (talk) 19:54, 9 October 2017 (UTC)

I don't succeed in getting clear what you mean with  "identify a series with the sequence of its partial sums". What do you mean here with "a series"?   According to Bourbaki and the Encyclopedia of Mathematics: "a pair of a sequence and its sum sequence". But a pair of two sequences is something else as a single sequence, so the consequences of your 'identify' remain mysterious to me.   On your wording of the 'Alternating series test'. Just as the symbolic form Σ a_i can have different meanings depending on the context, the word 'convergent' has to be interpreted depending on its context ('having a limit' or 'having a sum' = 'having partial sums with a limit'). Neither of the two seems to be applicable in your  "for a sequence to converge it is sufficient that its differences are alternating and monotonically decreasing to 0 in absolute values". -- Hesselp (talk) 00:18, 10 October 2017 (UTC)

Well, it seems I did my best already, and have nothing to add. Probably this is just another frustrating discussion. We fail to communicate. I understood that you dislike the "series" terminology and want to replace it with "sequences only" terminology. Finally I agree that this is possible, and show you some examples of the usual formulations translated into the new language (just to look, does it sounds good or not). And you miss my point completely and write that I cannot treat a series as a singe sequence and a pair simultaneously! Quite a miscommunication. And in addition, I see that you violate your topic ban, thus probably you'll be banned completely, and this discussion will stop anyway. Not a happy end. Boris Tsirelson (talk) 05:33, 10 October 2017 (UTC)

My topic ban concerns Series (mathematics) and its talk page (see here). So I suppose that my edits on Cesàro summation did not violate the ban.
Yes, we fail to communicate on a central point. It is as it is.  Anyway, I want to say that I'm very, very grateful for all your patience with me. And that you helped me to get my views on the subject even more concrete, and to find more compact and to-the-point wordings.
Although unfortunately not enough.... It seems that I have to accept that most mathematicians are born with a belief in the existence of 'series(es)'. So that they can write "treat a series as a single sequence and as a pair" without a specification/clarification about what they intend to communicate with the word "series" in that phrase. I miss this gene. With regards. -- Hesselp (talk) 13:54, 10 October 2017 (UTC)

Now I guess, if someone will say you: "I completely agree with your position about series", you'll cray: "Stop saying nonsense! With my position about WHAT??"   :-)   Boris Tsirelson (talk) 15:49, 10 October 2017 (UTC)

That's your guess.  In fact there is, indeed, a chance - depending on the 'someone' - that I should try to check whether this someone uses the word 'series' in the same way as Cauchy cum suis. -- Hesselp (talk) 17:53, 10 October 2017 (UTC)

June 2017

Welcome to Wikipedia. Everyone is welcome to contribute constructively to the encyclopedia. However, talk pages are meant to be a record of a discussion; deleting or editing legitimate comments, as you did at User talk:Kudpung, is considered bad practice, even if you meant well. Even making spelling and grammatical corrections in others' comments is generally frowned upon, as it tends to irritate the users whose comments you are correcting. Take a look at the welcome page to learn more about contributing to this encyclopedia. Thank you. — O Fortuna ^{semper crescis, aut decrescis} 15:18, 23 June 2017 (UTC)

@O Fortuna ^{semper crescis, aut decrescis}   By mistake, at 15:01, 23 June 2017, I posted an edit in User talk:Kudpung not in the newest version of that page. I apologize for the possibility that this could have been seen as intentional. -- Hesselp (talk) 16:32, 24 June 2017 (UTC)

@Hesselp: No problems. I was half expecting Kudpung to bollock me for reverting you. But if you made an accidental edit, surely no probem. So we are all good, and not de-bollocked. Yet, anyway. And I agree, accidents happen, And, I hope, are resolved. Happy editing here! Take care! — O Fortuna ^{semper crescis, aut decrescis} 16:38, 24 June 2017 (UTC)

Hesselp, please stop your incompetent edits to series related articles. I recently had to revert your edit to Cesaro summation, since you obviously have no idea what that is. Sławomir Biały (talk) 00:26, 10 October 2017 (UTC)

ANI notice

There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. D.Lazard (talk) 16:57, 30 October 2017 (UTC)

Topic Ban

I've closed your ANI thread with consensus to indefinitely topic ban you from all articles on or related to mathematical series. Effective immediately, if you are found to be editing any mathematical series article, its talk page, or anything else remotely related to mathematical series you will be blocked. A record of your ban has been noted at Wikipedia:Editing restrictions/Placed by the Wikipedia community, and you will remained topic banned until the community elects to let you edit the articles again or until the arbitration committee takes up the case. TomStar81 (Talk) 22:53, 7 November 2017 (UTC)

There is currently a discussion at Wikipedia:Administrators' noticeboard/Incidents regarding an issue with which you may have been involved. Sławomir Biały (talk) 23:09, 22 November 2017 (UTC)

E

Please stop trying to add material at e (mathematical constant). Consensus was clearly against the edits you wanted to make back in November when you stopped editing. I thought you understood then, but it would seem you don't. –Deacon Vorbis (carbon • videos) 15:57, 27 April 2018 (UTC)