Talk:Series (mathematics)

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Sum Of Geometric Series[edit]

I have taken this from a math textbook, but i dont want to post it until i find the copyright information, can someone confirm that this is correct?

"The sum of a finite geometric series is . If this finite sum S of n approaches a number L as n to infinity, the series is said to be convergent and converges to L and L is the sum of the infinite geometric series.

Thm: Sum of an Infinite Geometric Series:

    If the absolute value of r is less than one, the sum of the infinite geometric series  is   —Preceding unsigned comment added by Dandiggs (talkcontribs) 21:07, 31 January 2008 (UTC) 

Properties of Series[edit]

I think that there should be a section on the properties of series, such as multipication of series and commutativity of multiplied series. Lore aura (talk) 10:07, 28 April 2008 (UTyC) —Preceding unsigned comment added by Lore aura (talkcontribs) 10:05, 28 April 2008 (UTC)

Partial sum[edit]

What is a partial sum? Partial sum is a redirect to this page, even though it is linked to from various other math pages. There is no partial sum subsection in this article. --Cryptic C62 · Talk 02:24, 25 May 2008 (UTC)

In response to this question, I've improved the definition and rejigged the first bit of the page. Still needs a lot of work though! SetaLyas (talk) 02:00, 29 December 2008 (UTC)

Yea, I still have no idea what a partial sum is. McBrayn (talk) 15:10, 16 April 2009 (UTC)

From the article:
Basic properties
Given an infinite sequence of real numbers , define
Call the partial sum to N of the sequence , or partial sum of the series.
What more should one say? --Bdmy (talk) 21:36, 16 April 2009 (UTC)


Remainder term redirects here but there is no introduction to the concept of remainder in infinite series on this page. -- (talk) 19:24, 5 May 2009 (UTC)

Indian Mathematics[edit]

The section on Kerala needs to be rewritten as it incorrectly implies that the Kerala school made a significant contribution that was built upon by others and worse implies that Gregory used this work.Xp fun (talk) 21:01, 15 August 2009 (UTC)

Can you tell us more accurately what happened? JamesBWatson (talk) 09:55, 20 August 2009 (UTC)
I'll try, there is a systematic list of articles which have been modified some time ago to include claims that this Kerala school had invented the technique or concept centuries before the generally accepted mathematicians or physicists.
The idea behind this is in a couple of books cited in each article which alleges (not having read the book) that Madhava on the Kerala school (or his disciples) had discovered these ideas and through trade and commerce the ideas came to western mathematicians.
Now there are several websites which site these same couple of books, and these websites are used as additional links in citations creating a circular web of authority. Anyone reading any of these updates would probably check the links, see that they appear to research actual texts, and stop there. Only digging deeper do we see that there is no further original research than the first author.


First, the source articles:

Articles potentially tainted (Found via search of "madhava or Kerala")

... the list goes on, more exhaustive search will be required. List of supplied references

Cited Article Comment Citation
Mathematical_analysis#cite_ref-4 Madhava of Sangamagrama, regarded by some as the "founder of mathematical analysis". G. G. Joseph (1991). The crest of the peacock, London
History_of_science#cite_ref-15 In particular, Madhava of Sangamagrama is considered the "founder of mathematical analysis" George G. Joseph (1991). The crest of the peacock. London.
History_of_trigonometry#cite_ref-19 O'Connor and Robertson (2000)
History_of_trigonometry#cite_ref-20 Pearce (2002)
James_Gregory_(mathematician) Under See also is a link "Possible transmission of Kerala mathematics to Europe"
"In 1671, or perhaps earlier, he rediscovered the theorem that 14th century Indian mathematician..."
no citations at all
Mean_value_theorem#cite_ref-1 probably least biased reference I've found so far J. J. O'Connor and E. F. Robertson (2000). [[1]]

Ok, lets take that last one: O'Connor and Robertson. Actually, the site is a mirror of the MacTutor archive located at [[2]]

From there is a link to the interesting biography of Madhava [[3]]

And from there is the list of references: [[4]]

And Finally: at the top of the list: G G Joseph, The crest of the peacock (London, 1991)

I'm not disputing whether or not Madhava and his disciples did interesting things with geometry, nor whether the Mayan, Egyptian, or Native plains people of the Americas, had also discovered fascinating relations in nature. I'm objecting to the idea that this has had any relevance to the furthering of knowledge by the currently aknowledged authors of these ideas. Am I nuts here or are we witnessing an overzealous patriot trying to boost his/her country's esteem?Xp fun (talk) 18:16, 4 September 2009 (UTC)


Hi. Would it be possible at the beginning of the article to explain the sigma notation? I.e. what the small figures at the top and bottom of the sigma represent? I think that an introductory textbook would do this, and it would be helpful to many maths learners. Thanks for considering it. Itsmejudith (talk) 18:17, 11 November 2009 (UTC)


What difference between a "series" and a "sum of a sequence"? What is a "sum of a series"? What difference between the "sum of a sequence" and "sum of a series"? — Preceding unsigned comment added by (talk) 12:27, 15 June 2012 (UTC)

Read the article sequence to see that sums are not required. Further, a sequence may not converge to a limit. Next read partial sum. A sequence does not have a sum, but perhaps has a limit.Rgdboer (talk) 22:33, 18 July 2013 (UTC)

finite infinites[edit]

What about e.g. S = 1 + 10 + 100 + 1000 + ...
Most stupid people will tell you that it is infinity, it diverges, but I think, it is not: it's -1/9 (talk) 01:30, 28 August 2012 (UTC) - Nur weil ich verrückt bin, heißt das noch lange nicht, dass ich deswegen falsch liege.²³

Perhaps you're thinking of something like this? Isheden (talk) 08:29, 18 July 2013 (UTC)
10 S = S - 1 implies S = -1/9, very nice. So the message is that some calculations are only allowed if the series converges. Bob.v.R (talk) 02:03, 16 April 2017 (UTC)

Open problem?[edit]

I don't see the series

mentioned in the article. Is it still true that calculating the sum is an open problem? [5] Isheden (talk) 08:36, 18 July 2013 (UTC)

After some time I found a complete article on this sum: Apéry's constant Isheden (talk) 09:23, 19 July 2013 (UTC)

Tag "image requested"[edit]

I have removed the tag "image requested". I think that an image would be a good thing for this article. But, like for many mathematical articles, it is not clear which kind of image would improve the article. Therefore inserting the tag without suggesting the nature of the image that is requested is a non-constructive edit. D.Lazard (talk) 11:38, 20 September 2013 (UTC)


What is the indexed number n called? Is it the "summation variable"? —Kri (talk) 12:38, 18 October 2014 (UTC)

This is not incorrect, but "summation index" is more frequently used. D.Lazard (talk) 14:08, 18 October 2014 (UTC)
Our summation article says says "index of summation". --Mark viking (talk) 16:42, 18 October 2014 (UTC)
Sometimes it is not used as an index, though. Can it stille be referred to as a summation index? E.g. . —Kri (talk) 15:34, 19 October 2014 (UTC)
Yes, it can be referred to as a "summation index". Be care that in , n is not really a variable in the sense that it cannot be substituted by a value. It would better be called a "placeholder", as n may be replaced by any symbol without changing the meaning and the value of the expression. Sure that "index" often means subscript, but, in mathematics, it may also mean "discrete variable", as in indexed family. D.Lazard (talk) 16:39, 19 October 2014 (UTC)
Sure it is a variable; it's just a scoped variable and hence cannot be controlled from outside of the series. Hm, I don't know if I would still call it a summation index if it is not actually an index. —Kri (talk) 19:29, 20 October 2014 (UTC)

Indexed by natural numbers or non-negative integers?[edit]

I see the article starts series both at 1 and at 0 without any mention as to why it doesn't matter. If it is indexed by the natural numbers shouldn't start with 1 instead of 0? — Preceding unsigned comment added by (talk) 00:17, 21 July 2015 (UTC)

Alternative for the unconceivable:   [...] series is [...] the ordered formal sum [...][edit]

No simple clear description can be found for the mathematical object meant by the defining phrase "an ordered formal sum of an infinite number of terms". Yet the word 'series' is frequently used in mathematical texts, so the question remains: what is in fact communicated by this word?   I'll give my answer; please comment on it.

The word 'series', as well as the word 'sequence', refers to mappings on the natural numbers (the Peano structure); the words are synonyms as far as their mathematical content is considered.
The choice for the word 'series' is often made to announce or to emphazise that something will be said about the limit of the partial sums of some mapping on N: concerning the existence of this limit (with words as convergent/divergent/to converge/to diverge), or concerning this limit as a number (the sum of the mapping on N under consideration).
Moreover, in case the word 'series' is used for a mapping on N (say: a), as a notation for this mapping the commas form
a1, a2, a3, ... (, ai , ...)   is often replaced by the plus-signs form   a1 + a2 + a3 + ... (+ ai + ...)   or the sigma form   Σi =1,2,... ai   .
Two remarks:
1. The plus-signs form and the sigma form are also used for the sum of a (and sometimes as well as for the sequence of partial sums of a).
2. In almost all modern texts the words convergent/divergent/to converge/to diverge, in combination with the word 'sequence', apply to the terms, and not to the partial sums.   In some older texts (mostly 19th century, following Cauchy) the verbs are used only in combination with 'sequence', and the adjectives only with 'series'; the word 'convergence' doesn't occur. See Bradley R.E., Sandifer C.E., 2009, Cauchy's Cours d'analyse - An Annotated Translation

(p.85) We call a series an indefinite sequence of quantities,
u0, u1, u2, u3, ··· ,
which follow from one another according to a determined law.
(p.86) Following the principles established above, in order that the series
u0, u1, u2, ···, un, un+1, ···
be convergent, it is necessary and it suffices that increasing values of n make the sum
sn = u0 + u1 + u2 + ··· un-1
converge indefinitely towards a fixed limit s.

--Hesselp (talk) 14:57, 19 January 2016 (UTC)

I agree that "series" and "sequence" are fundamentally the same concept. However, we need to remember that articles like this are supposed to talk to as general an audience as possible and not just to mathematicians. I don't think these ideas will improve the article, especially not in the lead. McKay (talk) 02:42, 20 January 2016 (UTC)
"The same concept". Okay. So why should we go on with a Wikipedia article strongly suggesting (lying?) that 'series' and 'sequence' stand for different mathematical things? Cannot we find simple words to say that in certain situations 'sequence' is frequently replaced by 'series' (and in that case: 'summable' by 'convergent', and the comma notation by the plus-signs or the sigma notation)?
The present text starts with "This article is about infinite sums." Is it clear for a general audience what is meant with "sums that aren't normal sums"? --Hesselp (talk) 16:14, 20 January 2016 (UTC)
Firstly the sentence "This article is about infinite sums" is not a part of the article, it belongs to a disambiguation hat note.
"The same concept". No. Although in common language "series" and "sequence" are almost synonymous, in mathematics, they refer to concepts that are different although strongly related (to each series one may associate the sequence of its partial sums, as well as the sequence of its terms, and to each sequence one may associate the series whose terms are the differences of successive elements). This is the reason for which I have moved "In mathematics" in the article. To see that series and sequences are different concepts, it suffices to consider the product: The product of two sequences is obtained by multiplying together the terms that have the same index. On the other hand, the product of two series is a series that has a completely different definition; it is chosen in order that, if the series are (absolutely) convergent, the sum of the series product is the product of the sum of the series factors. D.Lazard (talk) 18:23, 20 January 2016 (UTC)
@D.Lazard. 1. The very first sentence " not a part of the article".   POV?
2. Your pretended strong relation between a sequence and a 'series', doesn't clarify what you mean with 'series'. We wait for a better explanation than the mysterious "an ordered formal sum of an infinite number of terms".
3. The Cauchy product of two sequences is defined in exactly the same way as it is for two 'series'. You agree?
4. See Cauchy's original Cours d'Analyse in French, p.123 and tell us where he went wrong. --Hesselp (talk) 21:35, 20 January 2016 (UTC)
1. See WP:HATNOTE. These aren't considered part of the article. They are disambiguation so that readers can navigate between articles when their titles are ambiguous. (Thus "disambiguation"). 2. Series form the total algebra over the monoid of natural numbers. If you equip the set of sequences with the Cauchy product, then the set of sequences with this additional structure can be identified with the set of series. But it is not right to say that, therefore, sequences and series mean the same thing. They are equipped with different structures. (Compare the differences between regarded as a vector space, a topological space, an inner product space. It's wrong to say that they're all the same thing.) Sławomir
13:34, 21 January 2016 (UTC)
To Slawekb, thanks for your comments.
Ad 1. On your "These aren't considered part of the article.":   I know, that's why I wrote (16:14 20 Januari 2016) "The present text starts with ....".
Ad 2. Please, could you transform your "Series form the total algebra over the monoid of natural numbers." into a wording for the Wikipedia audience? --Hesselp (talk) 15:58, 21 January 2016 (UTC)
I don't care much for the present lead much. Why is there so much bold ("series" is bold twice, each of "infinite sequences and series" and "finite sequences and series" and "infinite series" is in bold)? Why does the second paragraph begin "In mathematics..."? Is the subject of the first paragraph not also mathematics? In fact, why is the first paragraph there at all? The entire article is about infinite series rather than finite series. Sławomir
13:31, 20 January 2016 (UTC)

How to denote a sequence?[edit]

To, and ... .  On Februari 28 2017, you changed {...} into (...) at several places. I understand your argument (a sequence is a mapping, not a set), but I see your solution as insufficient. For without any harm, you can do without braces/parentheses at all, and without any index symbol as well.   A sequence is defined as a mapping on the set of naturals, so label them with a single letter. Just as people mostly do with mappings/functions with other sets as domain: f, g, F, G, ... .
When there is a risk of confusion you can write "sequence s", "sequence S"  in stead of just "s" or "S".
Who has objections? (Yes, I know the index is tradition, but it is superfluous and therefore disturbing.)
In the Definition section, three lines after "More generally ..."  I read:
    the function is a sequence denoted by .
I count three different notations for the same domain--function (sequence), four lines later a fourth version - - is used.
Last remark: It's not correct to say that sequences ( and , or simply and ) are subsets of semigroup . -- Hesselp (talk) 19:43, 10 April 2017 (UTC)

Index sets as generalization of N (subsection Definition)[edit]

For me it is impossible to find any information in the second part of subsection 'Definition'- after 'More generally....'.
The text seems to suggest that the notion of "series" (whatever that is ...) can be extended from something associated with sequences (mappings on the set of naturals) to a comparable 'something' associated with mappings on more general index sets.  But nothing is said about how such generalized mappings can be transformed into a limit number .  Is it possible to generalize the tric with the 'partial sums'? This index sets has to be countable? No reference is given. (The present text is composed by Chetrasho July 27, 2011).
I propose to skip the text from 'More generally' until 'Convergent series'.   Any objections? -- Hesselp (talk) 13:19, 11 April 2017 (UTC)

That would not be a good idea. Some of your these questions are answered in the section devoted to more general index sets. The entire article is rather poor on providing citations, so removal of material because it is unreferenced would decimate this article. Perhaps tagging the appropriate section with a lack of citations tag would be more useful.--Bill Cherowitzo (talk) 17:09, 11 April 2017 (UTC)
Hello Bill Cherowitzo.   You are right, the two questions are answered in the final section of the article.
But I persist that the description of the notion named series becomes even more unclear by adding six sentences (the greater part of the Definition section) on a generalization that will be unknown to most readers.
Moreover, the correlation between the position of this notion connected with sequences, and its position connected with mappings on an index set, is not very strong. For:
In (elementary) calculus two different symbolic forms (both named 'series') are used, expressing the relation between a sequence and its 'sum'. One of them, the plusses-bullets form  cannot be used in the generalized situation. And the other one, the capital-sigma form needs adaption ( instead of or or or ).
The absence of relevant information in this six sentences is not undone by a 'lack of information tag'. And skipping this sentences I cannot see as a "removal of [relevant] material". -- Hesselp (talk) 21:44, 11 April 2017 (UTC)
I've reconsidered this and agree that this discussion of summations doesn't belong in the series definition section. I've moved it to the appropriate section and tagged that section. Summation notation for uncountable indexing sets can be defined to make sense, but calling these things "series" may be problematic. A narrower concept of generalized series fields does exist in the literature, and this might be germane to the article.--Bill Cherowitzo (talk) 18:32, 12 April 2017 (UTC)
I agree with the removal of mentioning generalized index sets from the Definition section. But I still don't see which relevant information is added by the last two sentences in the present version of this section, to what is in the first three.
And I repete my 'Last remark' 10 April 2017: sequences (mappings on N) are not subsets of 'semigroup G '. -- Hesselp (talk) 06:36, 13 April 2017 (UTC)
Hopefully I have clarified the relationship and have removed the offending statement. --Bill Cherowitzo (talk) 16:16, 13 April 2017 (UTC)

Again on the Definition section[edit]

Yesterday's (13 April 2017) reduction in this section is an improvement, yes. Now this shorter version makes it easier to explain my objection to its central message. I paraphrase this message in the next four lines:
1. For any sequence is defined a
2. associated series Σ (defined as: an ordered "element of the free abelian group with a given set as basis" - the link says).
3. To series Σ is associated the
4. sequence of the partial sums of .
Why in line 2 an 3 a detour via a double 'association'(?) with something named 'series'? Is the meaning of that word clearly explained in this way to a reader? I don't think so.   I'm working on a text that starts with:
"In mathematics the word series is primarily used for expressions of a certain kind, denoting numbers (or functions). Secondly"
I plan to post this within a few days. -- Hesselp (talk) 13:32, 14 April 2017 (UTC)

I would be careful about this. This section is supposed to give a formal definition of series, the informal definition can already be found in the lead. The terminology here is fairly standard and any large deviation would require citations in reliable sources to prevent it from being immediately removed. --Bill Cherowitzo (talk) 19:08, 14 April 2017 (UTC)

Who can tell me how to find out whether or not a given "ordered element of the free abelian group with a given set as basis" has 100 as its sum? Who can mention a 'reliable source' where the answer can be found?
Why should this mysterious serieses be introduced at all, in a situation where it's completely clear what it means that a given sequence has 100 as its sum. I cannot find a motivation for this in a 'reliable source' mentioned in the present article.
So let's skip this humbug (excusez le mot).

About an eventual 'immediate removal': Should I have to expect that a majority in the Wiki community will support removing a serious attempt to describe in which way (ways!) the word 'series' is used in most existing mathematical texts. And replace a version including a 'definition' which has nothing to do with the way this word is used in practice; only because the wording has some resemblance with meaningless wordings that can be found in (yes, quite a lot of) textbooks.
In the present 'definition' of series the words 'formal sum' are linked to a text on Free abelian groups. Can this be seen as a 'reliable source' for a reader who wants to know what could be meant by 'formal sum'?  Wikipedia is not open for attempts to improve this? -- Hesselp (talk) 23:13, 14 April 2017 (UTC)

Comments on changes in the Definition section[edit]

Line 3, quotation: "Summation denote a series, ..."
A notation to denote an expression ??  Sounds strange (first sentence says: series = expression of certain kind).

Line 4, quotation: "Series are formal sums, meaning... by plus signs),"
I can read this as: "The word 'sum' has different meanings, but the combination 'formal sum' is a substitute for 'series' (being forms consisting of sequence elements/terms separated by plus signs)".   Correct?, this is what is meant?
But "Series are formal sums" seems to communicate not the same as " 'series' is synonym with 'formal sum' ".

Line 4-bis, quotation: "these objects are defined in terms of their form"
With 'these objects' will be meant: 'these expressions (as shown in the first sentence)', I suppose. But then I miss the sense of this clause. An expression IS a form, and don't has to be defined (or described?) in terms OF its form.

Line 6. Properties of expressions? and operations defined on expressions? This regards operations as enlarging, or changing into bold face, or ...?

Line 7. "...convergence of a series". In other words: "convergence of a certain expression"?   I'm lost.

I'll show an alternative. -- Hesselp (talk) 15:14, 16 April 2017 (UTC)

Three proposals for adaptations in the Definition section
I. Note 3 in the present text, saying "...a more abstract given in....", should be removed.
For it doesn't have any sense to refer to a  'more abstract definition'  of
    an expression of the form ,  labeled with the name 'series'.
There is not a  'less abstract definition'  of this kind of expression either. Only a description.

II. A more direct formulation of the third sentence in this section is:
"A series is also called formal sum, for a series expression has a well-defined form with plus signs."

III. Remarks on the 'usefullness' and the 'fundamental property' of such  expressions of the form  , shouldn't be included in a definition section. -- Hesselp (talk) 13:07, 17 April 2017 (UTC)

Elaborating Lazard's description of 'series' as an expression[edit]

I'm pleased to see that Lazard (Febr.14, 2017, line 4) describes the meaning of the word series as an expression of a certain type.   Less clear (or better: mysterious) is the remark: "obtained by adding together all terms of the associated sequence"; what could be meant by "adding together"? What kind of action should be performed, by who, on which occasion, to obtain / create an expression of the intended kind?
More remarks on the present text of the article, in this Talk page: 15:14 16 April 2017.
To get things clear, I propose to start this article in about the following way:

[sources: Cauchy, see p.123 and p.2 quantité C.L.B. Susler, 1828, Susler, S.92, Carl Itzigsohn, 1885, Bradley/Sandifer, 2009 ]

[More sources on the problem with 'series' in books/publications by: professor H. Von Mangoldt, E.J. Dijksterhuis, H.B.A. Bockwinkel, professor N.G. de Bruijn, professor A.C.M. van Rooij, professor D.A. Quadling, Mike Spivack Spivak, H.N. Pot; links have to be added. Several of this sources are written in Dutch.] -- Hesselp (talk) 15:38, 16 April 2017 (UTC)

H o w   t o   r e d u c e   c o n f u s i o n
The best thing to do is:  Stop using the word 'series' at all, and say:
(absolute) summable sequence and summability tests,  in stead of: (absolute) convergent series and convergence tests .
Second best is: inform students and readers of Wikipedia about the historical source of the confusion. Let them understand that the existence of any definable notion 'series' (different from 'sequence') is a wide-spread misconception. And train them to interprete (absolute) convergent series as nothing else as summable sequence. -- Hesselp (talk) 13:12, 17 April 2017 (UTC)

Series being a fundamental concept in mathematics, nobody can stop using series. All Hesselp's considerations show that he has not understood what a series is (in mathematics). So his propositions for rewriting the definition of a series is WP:OR, and have not their place in Wikipedia. Nevertheless, section "Definition" was poorly and pedantically written, with a confusing emphasis on sequences. Thus, I have rewritten this section, hoping that readers such as Hesselp, will be less confused. D.Lazard (talk) 15:12, 17 April 2017 (UTC)
Answer to D.Lazard: Thank you for contributing to the search for the best way to describe what is meant with the word 'series' in texts on mathematics(calculus). I saw some points in your rewriting of the Definition section which I can see as improvements. But there are some problems left:
1.   Rewording the first sentence more close to the usual way as definition of 'infinite series / series',  I get:
An infinite sum is called series or infinite series if represented by an expression of the form: . . .
This paraphrasing is correct?
Please add an explanation of what you mean by 'infinite sum'.  And tell how a blind person can decide whether or not he is allowed to say 'series' to such an infinite sum, as he cannot see the form of the representation.
2.   In the third sentence 'summation notation' is introduced, showing a 'capital-sigma' form, followed by an equal sign and a 'plusses-bullets' form. Why two different forms to illustrate the 'summation notation'?
3.   Please explain what you mean with 'formal sum' (fourth sentence). See this discussion. And the same question for 'summation' at the end of that sentence.
4.   Your seventh sentence end with "...the convergence of a series". Do you really mean to define "the convergence of an expression(of a certain type)" ?
5.   Finally, I'ld like to see an explanation of the clause "the expression obtained by adding all those [an infinite number of] terms together" (fifth sentence in the intro). I don't see how the activity of 'adding' (of infinite many terms!) can have an 'expression' as result. -- Hesselp (talk) 20:01, 17 April 2017 (UTC)

Reaction to D.Lazard. On his remarks concerning my 'lacking understanding' of what a series IS, and my proposals for rewriting THE definition of a series. (The 'IS' and 'THE' referring to Lazard's personal POV.)
Cauchy used 'série' in his publications according to the definition:
    "On appelle série une suite indefinite de quantités (= nombres réelles)".  See 1821 Cours d'Analyse p.123,2
You agree that in modern English this reads as "An infinite sequence of reals is called series." ?  A clear definition?
(Maybe later on Cauchy used the same word to denote sequences of complex numbers as well.)
Probably by his choice for "convergente" naming the property now called "Fr: sommable / En: summable", a permanent confusion arose.
Numerous alternative attempts to define 'series' can be found, all of them denying Cauchy's distinction between 'converger / to converge'  versus  'convergente / convergent'. This attempts can be quite diverse, see for instance Bourbaki's: "a pair of sequences (an), (sn)".  None of this attempts is satisfying, for all of them have undefined clauses as  'infinite sum',  'formal sum',  'obtained by adding all those terms together',  'if we try to add the terms of...we get...'  'summation'.
The word 'series' is used by mathematicians, yes!  (Although there are complete textbooks on calculus, intentionally totally avoiding this word.)  So readers of Wikipedia should be offered a clear explanation of possible interpretations of this word when occurring in a mathematical text. A single, cripple, 'definition' is not enough. (My personal POV.) -- Hesselp (talk) 13:22, 18 April 2017 (UTC)

Lacking section: Operations on series[edit]

After having rewritten the section "Definition", I have remarked that the operations on series are not defined here. Thus a section must be written for describing how defining addition, multiplication, multiplicative inverse (if the first term is invertible), derivative and antiderivative for series, and stating that if the argument of the operations are convergent, the same is true for the result, and, in that case, the sum of the result of an operation on series is equal to the result of the same operation applied to the sums of the input series. I have not the time to write this section. Can someone do that? D.Lazard (talk) 15:46, 17 April 2017 (UTC)

R e d u c t i o n   o f   s u m s   a n d     p r o d u c t s
A sum of two numbers given in series representation,
a product of two numbers given in series representation, and
a product of two numbers, one of them given in series representation,
can be reduced according to:
    (sequence or sequence summable)
The same applies for functions instead of numbers. -- Hesselp (talk) 22:17, 17 April 2017 (UTC) -- Hesselp (talk) 13:22, 18 April 2017 (UTC)

Motivation for partly substituting the text of "Series (mathematics)"[edit]

The present text strongly suggests that there is only one correct interpretation of what is meant by the word 'series' in mathematical texts. That is that the word 'series' is the name for a certain idea / notion / conception / entity. But what IS "it"?
"It" is NOT a number.
"It" is NOT a sequence (= a mapping on N)
"It" is NOT an expression (for the present text says: "a series is represented by an expression)
"It" is NOT a function.
"It" is 'associated' (what's that?) with a sequence.   "It" is sometimes 'associated' with a value.
"It" has terms and partial sums.
"It" can have a limit, a value, a sum.
"It" can be geometric, arithmetic, harmonic, alterating, converging, diverging, absolute converging, and more.

What's in fact the content of this black "it"-box?   It seems to be empty.
I'm going to replace this unsatisfactory text by an alternative introduction. Chiefly identical with what was shown in this Talk page here, 18 April 2017. The only reaction on it was the remark that "Hesselp doesn't understand what A SERIES IS (in mathematics)".  I agree with that. --Hesselp (talk) 22:05, 24 April 2017 (UTC)

Comment on the undoing of revert 03:09, 25 April 2017[edit]

Undocumented? POV? : see Talk page from 14 April 2017 on. There should be mentioned, line by line, where and why the text of the reverted alternative is seen as not a correct description of how the word 'series' is used in mathematical (calculus) texts in practice. --

Please read WP:BRD: When an editor that want to change an article, and has been reverted, it must not start WP:edit warring. Instead, he must start a discussion on the talk page for trying to convince the other editors that his edits improve the article, and for trying to reach a WP:consensus on the best version. D.Lazard (talk) 10:31, 25 April 2017 (UTC)
I acknowledge that you have started a discussion, to which three editors (including myself) have participated. All have edited the article for fixing some issues that were revealed by your edits and comments. This shows that they have really tried to understand tour point of view. But, it is clear that none agree with your proposed change of the article. There is thus a consensus against your version (which has been reverted by two of them). D.Lazard (talk) 10:54, 25 April 2017 (UTC)
@D.Lazard. Your 'edit summary' on 25 April 2017 says: "Editor's personal opinion not supported by sources". Without specifying the lines in the reverted text, in which you found a 'personal opinion', and in which more sources are needed according to you. In your remarks on this Talk page, you don't say anything more than that D.Lazard and Wcherowi don't agree with the proposed changes. Nothing on the discussion points on this page, posed on 20:01, 17 April 2017(UTC) and on 22:05, 24 April 2017(UTC). That's not taking part in the discussion as meant in WP:BRD, so your revert was not in accordance with that directive.
One more effort to start discussion.
The present text starts with:   "A series is, informally speaking, the sum of the terms of an infinite sequence."   The terms are numbers, and the sum of numbers is again a number. But: no mathematician uses the word 'series' as a synonyme for 'number'.
Please explain why you prefer this first sentence over the alternative:   "In mathematics (calculus), the word series is primarily used as adjective specifying a certain kind of expressions denoting numbers (or functions)."  (Omit  "as adjective"  if you want.) --Hesselp (talk) 23:34, 26 April 2017 (UTC)

Neither I nor any other editor is obligated to refute your arguments, just pointing out that your edits are not supported by citations to reliable secondary sources is sufficient for their removal. You seem to be under the impression that Wikipedia is an appropriate place to publish your views, but it is not. We have very strong guidelines against what you are attempting (WP:NOR and Wikipedia:SYNTHNOT) and beyond that, Wikipedia is not the place to be righting all the wrongs in the world. If you want Wikipedia to represent your point of view, then get it published in some reliable venue and after it is vetted by the mathematical community we will consider it for inclusion here. None of this, by the way, says anything about the merits of your arguments, some points of which I actually agree with. It is your profound misunderstanding of what Wikipedia is all about that is making some editors antagonistic in this situation.--Bill Cherowitzo (talk) 17:16, 25 April 2017 (UTC)

@Wcherowi. I repeat your sentences and insert my comments:
Neither I nor any other editor is obligated to refute your arguments,
Okay, no one is obligated to write any word or sentence on this Talk page. But when someone makes a revert, I expect a clear motivation on why text B is seen to be of higher value for Wikipedia readers than text A. A motivation, taking into account the arguments that are shown before (that's not the same as 'refuting these arguments', for maybe that could be a difficult task in some cases).
just pointing out that your edits are not supported by citations to reliable secondary sources is sufficient for their removal.
I suppose you mean: text A is "not enough supported by ..." (I'll give a list below). Here the question comes up whether or not text B is more / better supported by this kind of sources. "The sources of this section remain unclear"  I read on top of subsection 'Definition' in (the present) text B. That's in line with the impossibility to find any reference to a source, giving a non-contradictory description of the (supposed) notion named by the word 'series'.
You seem to be under the impression that Wikipedia is an appropriate place to publish your views, but it is not.
But what to do, in case one my 'views' coincide with what I consider as a possibility to improve an existing text?
We have very strong guidelines against what you are attempting (WP:NOR and Wikipedia:SYNTHNOT) and beyond that, Wikipedia is not the place to be righting all the wrongs in the world.
I'm attempting to bring into the article a better description of the (diverse) ways the word 'series' is used by mathematicians. Wikipedia guidelines are against that?
In WP:NOR I found (foot-note 1) that 'language' and 'readable online' are not limiting the required sources (on my list there are some in Dutch).   And in SYNTHNOT, line 5, is said: "After all, Wikipedia does not have firm rules."
If you want Wikipedia to represent your point of view, then get it published in some reliable venue and after it is vetted by the mathematical community we will consider it for inclusion here.
You can see the magazine of the Royal Dutch Mathematical Association as reliable? The article "No one can say what serieses are"; 2008 as representing 'my point of view'? And the review article 2009 as (partial) result of the screening by the mathematical community? (Togethe with an increased use of "summable sequence" in Dutch school-books. And in google-hits.)
None of this, by the way, says anything about the merits of your arguments, some points of which I actually agree with.
It is your profound misunderstanding of what Wikipedia is all about that is making some editors antagonistic in this situation.
"profound misunderstanding"?   It seems that your POV differs from mine, on this point.
Secondary sources supporting Hesselp's edits
- E.J. Dijksterhuis, book review (in Dutch), 1926-27 volume 3, no. 3-4, p.98-101: (paraphrased)  "To consider an infinite series as being an expression, seems to be less desirable."
- H.B.A. Bockwinkel, Integral calculus (in Dutch), 1932:   "The expression   u1 + u2 + u3 + ···   or  Σ1 un   is called a infinite series.  About what an author has in mind with respect to the meaning of this expressions, no information is given."
- P.G.J. Vredenduin, article (in Dutch) 1959 vol. 35, no. 2, p. 57-59:   "In Holland, in lessons on mathematics, normally no clear distinction is made between sequences and serieses."
- P.G.J. Vredenduin, article Sequence and series (in Dutch) 1967 pp.22-23:  "The problem how to define the meaning of the word 'series', is evaded by giving definitions for 'convergent series', 'sum of a convergent series' and 'divergent series',  but not for 'series' alone."
- M. Spivak, Calculus (editions 1967-2006):   "The statement that  {an}  is, or is not, summable is conventionally replaced by the statement that the series   Σn =1 an   does, or does not, converge. This terminology is somewhat peculiar, because………."
- N.G. de Bruijn, Printed text (in Dutch) of a series of lectures, 1978,  Language and structure of Mathematics:   "The way language is used with respect to serieses, is traditionally bad."
- H.N. Pot, article What serieses are, you cannot say(in Dutch), 2008
- A.C.M. van Rooij, article, review ofWhat serieses are, you cannot say (in Dutch), 2009:  "Instead of convergent serieses, you will have summable sequences, and everything is okay.  A bonus is that you don't use the word 'convergent' in two different ways." --Hesselp (talk) 23:34, 26 April 2017 (UTC)
- The authors of the texts behind the 40 000 google-hits with <summable sequence> and <summable sequences>. --Hesselp (talk) 22:52, 27 April 2017 (UTC)
It seems like you have identified a Dutch school of thought on this topic. This would probably be good for a paragraph in the article, but certainly not a rewrite.--Bill Cherowitzo (talk) 05:20, 27 April 2017 (UTC)
@Wcherowi.   Your remark on a 'Dutch school of thought', I cannot see as a way of participating in a discussion on the merits of certain wordings in version A compared with version B.
I'm amazed that an attempt to distinguish different meanings of the word 'series' in the vocabulary of mathematicians (instead of going on attempting to formulate what a series REALLY IS - handed down by God/Allah -), is judged as you do.
You don't give any reason why the fact that most of the cited sources are written in the language where I live, makes their content  c e r t a i n l y  not suited as base for a rewrite of the opening paragraphs (about 1/6 of the article).
Did you notice that all traditional wordings with 'series' are mentioned in the rewritten version? All of them with there meaning(s) carefully (I hope) explained.
I have not seen any reaction on the discussion points, presented at 20:01, 17 April(UTC) and at 22:05 24 April 2017(UTC). I understand that to make a revert by someone who is not taking part in the discussion on the merits of the two versions, is not in accordance with the directive in WP:BRD.   So I feel free to undo such reverts. And to go on trying to reach a version of this article in which the meanings of the word 'series' as used in mathematical texts, are described in a clear and unambiguous way. --Hesselp (talk) 22:52, 27 April 2017 (UTC)
I object to paraphrasing Hesselp's pov as a 'Dutch school of thought'. Several users on the Dutch wikipedia have again and again (and again) attempted to explain to Hesselp that wikipedia must inform the reader on facts and definitions as accepted and used by the (in this case) mathematical community, and that wikipedia is not meant to publish own research. As far as I know, the personal viewpoints of Hesselp do not represent a Dutch school of thought. Regards, Bob.v.R (talk) 00:44, 6 June 2017 (UTC)

Comment on the undoing of revert 10:23, 25 April 2017[edit]

The revert at 10:23, 25 April 2017 was made by someone who didn't participate in discussion on the merits of the competing versions. No reaction on the points raised at   20:01, 17 April 2017(UTC), and at   22:05, 24 April 2017(UTC). --Hesselp (talk) 22:52, 27 April 2017 (UTC)

You clearly don't have consensus to make these changes. Please wait until you have arrived at some kind of compromise rather than continuing to edit war. - MrOllie (talk) 23:06, 27 April 2017 (UTC)
@MrOllie.   "Clearly no consensus" ?   That's not very clear at all, for the 'reverters' didn't take part in any discussion on the merits of both versions (apart from "Undocumented POV pushing" and the like).
In more detail:   I extensively mentioned weak points and contradictions in the present text on how the meaning of the word 'series' is described. And showed how (according to me) this can be improved. None of the reverters contributed to discussion on this point. See:
- the draft version of the alternative (Elaborating D.Lazard's...)   15:38, 16 April 2017(UTC)
- the 'some problems left' (1 - 5)   20:01, 17 April 2017(UTC)
- the missing meaning of the "it" in a black box   22:05, 24 April 2017(UTC)
- the choice of the first sentence in the article, answering D.Lazard   23:34, 26 April 2017(UTC) .
The suggestion (Wcherowi) to add the alternative descriptions as a supplement, is an option but maybe not the most desirable.   Concrete arguments contra the present text being shown,  and concrete arguments contra the alternative being absent, I still see the undo of the revert(s) as sufficiently motivated and supported. --Hesselp (talk) 19:55, 28 April 2017 (UTC)
It's not a surprise it is difficult to get people to engage to your standards when you are posting walls of text on the talk page and attempting to rewrite so much in one go. I suggest you start small and propose one (small) paragraph at a time and see if you can build consensus. Contrary to what you seem to be saying here, that each 'reverter' did not engage with all of your many points, does not mean that you should edit war to keep your changes in while discussion continues. - MrOllie (talk) 21:40, 28 April 2017 (UTC)
@MrOllie.   On: "rewrite so much in one go":
The "so much" concerns one point: the way how to explain to readers the meaning(s) of the word 'series'. I don't see a way to split this issue in smaller paragraphs.   Although: the discussion here on Talk, can possibly proceed sentence by sentence. I proposed to D.Lazard just to start with sentence-1, see 23:34, 26 April 2017(UTC). No reaction.
On: "each 'reverter' did not engage with all of your many points":
Do I have to understand that this is an euphemism for "no one of the reverters did engage with any of the presented points"? I think so. --Hesselp (talk) 22:52, 28 April 2017 (UTC)
It wasn't intended that way, no. It's not really helpful to dwell on that either way: you need support to make these changes, and your present approach isn't gathering that support. - MrOllie (talk) 23:34, 28 April 2017 (UTC)
@D.Lazard.   I asked you (see edit summary), to write down here the arguments presented by 4 editors. In case you meant that I (Hesselp) am included in this 4, my request concerns the three 'non-supporters'. --Hesselp (talk) 22:52, 28 April 2017 (UTC)
@MrOllie. Only now I noticed that you wrote in your 'edit summary': "since this seems to be a cause of confusion, let us cite the definition we're giving in the lead". You ask me to copy here the first 10 sentences of 'my' intro?   Of which the first 3 probably can be discussed more or less separately. --Hesselp (talk) 23:12, 28 April 2017 (UTC)
No, that isn't remotely what I was saying. I wasn't asking you to do anything. I was providing a citation for the definition of a series used by the preexisting article. This is what English speakers mean when they use the term series. Perhaps Dutch speakers have a different definition (is this all just a problem of translation?) but I think that would be a matter for the Dutch language Wikipedia. - MrOllie (talk) 23:34, 28 April 2017 (UTC)

Mathematics, not religion[edit]

The present text presents in the intro plus subsection Definition, four different 'definitions', all of them using the wording:
"a series  IS  ..." .

1. (Intro, sentence 1)   "a series  IS  ... the sum of the terms of ..."
(Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym.)

2. (Intro, sent.5)   "The series of (associated with) a given sequence a   IS  the expression  a1+a2+a3+··· "
(The word 'series' used as the name of a mapping.)

3. (Definition, sent.1)   "a series  IS  an infinite sum, which is represented by a written symbolic expression of a certain type."
(It isn't clear whether or not the clause after the comma is part of the definition. 'IS' a series still an infinite sum, in situations where it is not represented by an expression of the intended form?)

4. (Definition, sent.6)   "series(pl)  ARE  elements of a total algebra of a ring over the monoid of natural numbers over the a commutative ring of the a's "
(The word 'series' as the name for elements of a certain structure; just as the word 'number' is used as the name for elements of another mathematical structure.   To which element in this 'definition' is referred by "the a's" ? )

In case it is accepted that the word 'series' has four different meanings in mathematics (is used in four different ways) the first part of the article headed by "Series" should be structured like:
a. The word 'series' is used as name/label for ......... .
b. The word 'series' is also used as name/label for ......... .
c. The word 'series' is used as name/label for .......... as well.
d. Moreover, sometimes the word 'series' is used as name/label for ......... .

The present text directs the reader to believe that there is ONE and only ONE sacred given-by-God-meaning of this word.
That's religion, not mathematics.
Do you think, Wcherowi, the summing up of different meanings is wrong?
Do you think, D.Lazard, the summing up of different meanings is wrong?
Do you think, MrOllie, the summing up of different meanings is wrong?
Do you think, Sławomir Biały, the summing up of different meanings is wrong?

One of the main reasons I see the present text as ready for improvement, I described earlier in

-- Hesselp (talk) 13:31, 30 April 2017 (UTC)

Addition to the list of notions NOT being a 'series' (an "it" ) :
    "It"  is NOT a part of Zermelo-Fraenkel set theory (NOT a part of the conventional foundations of mathematics).
As mentioned by Sławomir Biały, in edits 30 April, 14:40 and 14: 55 .
The mystery around the nature of "it" is growing even more. Who has ever 'seen this cat' ?
According to other people she clearly shows herself in the edit of 21:24, 18 28 April 2017(UTC).   What's wrong with that text? Line by line, please. --Hesselp (talk) 20:38, 30 April 2017 (UTC)

Yes, Series are not formally axiomatized by Zermelo-Frankel set theory. That contains a list of axioms that are taken to hold, none of which involves the concept of mathematical series. But series do exist in that theory, since it is possible to build models for them within that theory that are characterized by universal properties (e.g., as completions of algebras or abelian groups). So, in that sense, we "have seen this cat", meaning that we can construct a model of the universal characterization in ZFC. If you believe that no one has "seen" an infinite set, then that is not an appropriate speculation for this page. But also there are then much more mundane things one hasn't "seen" either, like numbers. Sławomir Biały (talk) 21:46, 30 April 2017 (UTC)
The basic definition is that of "an infinite summation", or a bunch of terms with plus signs placed between those terms. A series is therefore an infinite expression. In order to support the operations that most people would like to include, the terms should belong to the same ring, meaning that series of the same type can be added together and multiplied, as well as multiplied by elements of the ring. However, the concept of an "infinite expression" is not something that is axiomatized in Zermelo-Frankel set theory, so it is often useful to build a model of series that supports these operations in that theory. This is an interpretation of "series" in ZFC. So, to answer your question, there is one concept of series, that has multiple interpretations. I think this is something you really should try to understand before attempting to rewrite the article based on what others have pointed out is a failure to understand its subject. Sławomir Biały (talk) 14:30, 30 April 2017 (UTC)

(edit conflict) This article is about the concept of series in mathematics, and this talk page is about improving the content of the article, not for discussing personal opinions about the practice of mathematicians, and not for discussing mathematical standard terminology. It appears from your lengthy posts that you consider that there is no concept of a series, only a word with some meaning. This is a respectable opinion, but this is not mathematics, it is philosophy. Almost all mathematicians have an opposite opinion, which is usually expressed by saying that a series is a mathematical object. It appears that this concept is not a simple one, as it involves the concept of infinity, which was not well understood nor well accepted before the end of the 19th century (this make your citation of Cauchy irrelevant for discussing the modern terminology; note that he avoided carefully to talk about infinity). Presently, mathematicians agree on the concept of a series, but as usual for concepts that have many applications, the formal rigorous definition is too technical for being understood by beginners, but the full understanding of the formal definition is not really needed for manipulating and using the object, here a series. Similarly, you need not to know how a car is built for driving it, and you need not to understand the definition of the real numbers through Cauchy sequences for using them.
Some reasons for which you got few answers to your questions are the following:
  • Wikipedia is not a text book: I you have not the mathematical skills for understanding this article, take a standard textbook of calculus, and use it for learning the concept. This talk page is not the place for getting explanations, and editors usually prefer improving Wikipedia to giving you particular lessons.
  • As you have clearly a misconception of what is mathematics (not a religion, a science, which has been built, and continue to evolve through mathematicians work), you melt, in most of your posts, philosophical opinions, terminology considerations and misunderstanding of the object of this article. Thus discussing with you would be excessively time consuming.
  • You are asking (and WP:edit warring) for a major change, which consists, essentially, in changing the concept which is described. As the version that you insist to impose is not supported by most sources on the subject, this change is WP:OR, and, as such, may be reverted without any discussion.
D.Lazard (talk) 15:14, 30 April 2017 (UTC)
After my preceding post, Hesselp has modified his last post for asking What's wrong with that text? Line by line, please. Apparently, he has not read or not understood my last post, which was an answer to his post. For being clearer, every line of Hesselp's post is either wrong, or does not belong to this talk page or both. As I have explained this several times, no further answer is needed. D.Lazard (talk) 21:07, 30 April 2017 (UTC)
@D.Lazard.   Your post dated 15:14, 30 April 2017(UTC) starts referring  that this article is about the mathematical concept named 'series'.  Okay.
What I'm trying is:  to improve the description in the article of what is considered by mathematicians as the content of this concept ('mathematical object', as you say).   That's legal?
You wrote:  "a rigorous definition is too technical for being understood by beginners".   In my view a considerable reduction of this difficulties is furnished by skipping a number of generalizations of the original concept. By restricting (in the first part of the article) to serieses associated with real (or complex) sequences and real (or complex) functions. And with the plus sign only denoting the traditional addition.   You agree with this restriction?
I don't know whether this will be enough to make it possible to present a 'rigorous' definition of the (restricted) concept.  If not, be open/honest to the reader: say that a complete description is not presented here, and show references to other sources within or without Wikipedia.
And tell the reader that they can 'drive the car'  by reading  "series a1+a2+a3+ ··· is (not) convergent"
as   "sequence  (a1+···+an) n ≥1  converges" .
In words (suited to verbal communication):   "sequence a1, a2, a3, et cetera  is (not) summable" .   (Without the need to understand fully the deep rooted concept 'series'.)   Any objections? --Hesselp (talk) 10:36, 4 May 2017 (UTC)
I have indicated my objection to further edits without developing positive consensus. I reiterate my objection. Sławomir Biały (talk) 10:43, 4 May 2017 (UTC)

Critical remarks on the first twelve sentences of edit 30 April 2017, 14:59[edit]

1. (Sent.1)   "a series  IS  ... the sum of the terms of ..."
Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended.

2. (Sent.2)   "a series continues indefinitely"
What is meant by:   an indefinitely continuing 'sum of the terms of something' ?

3. (Sent.4)   "the value of a series"
What is meant by:   the value of a sum (a number) ?

4. (Sent.4)   "evaluation of a limit of something"
What's meant with this?
Is it true that a series doesn't have a value, without that limit being 'evaluated' ?
Is it always possible to 'evaluate' the limit of a sequence of terms ?

5. (Sent.5)   "the expression obtained by adding all those (an infinite number of) terms together"
A (symbolic, written) expression can be obtained by writing down some symbols using a pen or pencil (or using the keys of a keyboard). The task of adding an infinite number of terms is not feasible, so never any expression will be obtained.

6. (Sent.6)   "obtained by placing the terms side-by-side with pluses in between them.
This 'placing' sounds much better feasible. I miss the three centered dots ('bullets') at the right end.

7. (Sent.6)   "infinite expression"
I see 'series' and 'infinite sum' used as synonyms for 'infinite expression'. But what notion / mathematical object is denoted by this labels ? It must be a notion 'not being a part of the conventional foundations of mathematics'.  How many readers of this article are acquainted with this notion already by themselves?

8. (Sent.7)   "The infinite expression can be denoted ..." Such expressions mostly denote a number, a function or a sequence. But an expression denoting a expression sound very strange.

9. (Sent.9)   "two series of the same type"
I cannot find where is explained what is meant by:  'the type of that mysterious notion called series '.

10. (Sent. 8, 9, 10, 11, 12)
Is the (intended) information communicated by this five sentences really of enough importance to be incorporated in the 'introduction' ?

11. (First line after 'Definition') The twofold description of the meaning of the word 'series' (as sum, and as expression) causes - unnecessary? - complexity.
--Hesselp (talk) 23:13, 30 April 2017 (UTC)

A series is not a number. The lead does not even mention numbers, so I do not see how you are getting "Being the sum of numbers again a number, the words 'series' and 'number' are used as synonym. A few lines later it is said that this is not intended." Nevertheless, I think this cuts to the heart of your confusion. If I write "1+1", you would say that this finite sum is the number 2. But the expression "1+1" and the expression "2" are not the same expression. A "finite sum of numbers" is not merely another more complicated way to say "number". To be very precise, we should say that the expression "1+1" evaluates to the number "2". Now, for series (the subject of this article), the expressions still make sense, in that they can be formulated abstractly, but although the sum of a finite collection of numbers is again a number, the same is not true of the sum of infinite collections of numbers. For example, the expression , while it makes sense as a series, does not admit an interpretation as a number.
The sigma notation is a notation to refer to the infinite expression, just as we may introduce any other arbitrary name for it, like "x" or "Bob".
I have tried to clarify these things in the article, but these further comments do not make it seem likely that you are actually here to improve the article. Instead, your sarcastic wording ("the type of that mysterious notion called series") strongly suggests that you are not, in fact, going to even attempt to understand the subject of the article, do not appreciate the efforts of others to attempt to clarify these things, and indeed are quite possibly just trolling at this point. In any case, we have clearly reached the end of useful discussion. Accordingly, this is my final interaction with you. But I will continue to watch the article for your edits, and if you do this again, you can count me on the list of editors who strongly oppose this rewrite, and I will revert such edits without further discussion. Sławomir Biały (talk) 23:54, 30 April 2017 (UTC)

@Sławomir Biały, and other readers of this Talk page.
You write that you don't want to continue discussion;  it's your choice.  This doesn't prevent me from writing down my comments on what you put forward.
1)   About the 'mysterious' status of the notion/concept named 'series'.
I used the word 'mysterious' to refer in a short way to the   "it" is NOT a ....-list.  It was and is not meant as sarcastic.
On 30 April, 14:30 and 21:46 you're argumenting your view that  "there IS a (one) concept of series".   My hesitations to agree with you on this point, have to do with your formulations (wordings) like:
- it is often useful to build a model of series ...       - This is an "interpretation" of "series" ...       - Series are not formally axiomatized ...    - which includes the concept of mathematical series     - But series do exist ... to build a model of them.
Here you are suggesting every time that you have an a priori believe in the existence of a notion named 'series'.
There are believers, and there are non-believers.
2)   About "an expression denoting an expression".   To me this sounds still as strange as before.
You attempt to explain this by: "The sigma notation refers to the infinite expression".  But isn't it universally agreed that a sigma expression - in case it is not meaningless/void - denotes / refers to  a number (more general: a function) or a sequence?  Not an expression.
3)   About:  "The basic definition is ... a bunch of terms with plus signs placed between".
I see this as being very close to sentence 2-3 in my edit 21:24 28 April 2017(UTC):
        Symbolic forms like        and     or    expressing a number as the limit of the
        partial sums of sequence , are called series expression.  'Series expression' is often shortened to just 'series'.
I use the short notations for a mapping on N (a sequence) and as alternative for (avoiding problems with the first index). I know that this is not usual, so if this is seen as not desirable I don't persist.
My choice of wordings at some places has to do with my view on expressions in general: verbal expressions versus written expressions,  and written expressions using text versus written expressions using mathematical symbols.
4)   About:  "To be very precise, we should say that the expression "1+1" evaluates to the number "2" .
I think it's better to say:
the expressions "1+1" and "2" are equivalent (equi-valent = same value); or
the expression "1+1" can be rewritten as  "2" ; or
the expression "1+1" can be reduced to  "2" ; or
the standard form for the value of expression "1+1" is  "2" .
The meaning of "the evaluation of an expression" is not clear (to me). The expression  "e+π"  denotes (refers to) a certain (irrational) number. So the expression has a value. But the expression does not  'evaluate to a number' . --Hesselp (talk) 21:22, 1 May 2017 (UTC)
There is no semantic difference between saying that something "has the value" and "evaluates to". When you say that the expression "1+1" evaluates to/has value the number "2", you mean that the string of symbols "1+1" evaluates to the number 2. You are not saying that the expressions "1+1" and "2" are equivalent (the expression "2" is a numeral which also evaluates to the natural number 2; the number 2 is described, for example, in Zermelo-Frankel set theory as a particular set in that theory having nothing to do with the numeral expression "2". That numeral expression evaluates to the number 2 in ZFC.). (Also, one needs to be careful in using the word "equivalent" as if it meant "equi" + "valent". See equivalence relation. There are equivalence relations on the set of numerical expressions that are compatible with evaluation, but for which "1+1" and "2" are not equivalent expressions. Indeed, they are not identical expressions, and identity itself is an equivalence relation.) Similarly in your example "e+pi" is a string of symbols in a language, that evaluates to a certain irrational number when that language is interpreted in a model. Once you accept that "expressions" are not the same thing as "numbers", it is perfectly reasonable for there to exist expressions that do not correspond to numbers. The sigma notation for a series refers to the expression. Not the number. The series may admit an interpretation as a number, in which case we would say that the series evaluates to the number, in the same way that the expression "1+1" evaluates to the number 2, or the limit expression "" evaluates to/has value of 1 (the number). In any case, your own failure to accept that mathematical expressions are different things from numbers seems more and more like a matter of personal philosophy and taste than something of relevance to the article. You will just need to accept that series can be formalized in modern mathematics, if you're not actually willing to learn that mathematics. Sławomir Biały (talk) 12:46, 2 May 2017 (UTC)

To evaluate a given expression means ... ?[edit]

@Sławomir Biały.   Never in my life I've denied that mathematical expressions are totally different from numbers. You must have misunderstood me somewhere, I cannot trace back where this could have happened.
I agree with you on everything you wrote in the first seven sentences in 12:46, 2 May 2017(UTC)   (Until "The sigma notation for a series..."). About your sentences 8, 9, 10  I'm not sure. Maybe things become more clear from your judgment of the following statements a - h (true or false, or ...):

a)   the expression   e+π   evaluates to (= has as its value) the number   e+π

b)   the expression   1+1   evaluates to the number   1+1

c)   the expression   1+1   evaluates to the number   2

d)   the sigma expression   Σi =1 ai   evaluates to the infinite expression   a1+a2+a3+···

e) Provided that   limn→∞ (a1+ ··· +an)  exists,
   in other words   limn→∞ (a1+ ··· +an)  is a valid expression,
   in other words   sequence (an)  is summable,
      the infinite expression   a1+a2+a3+··· (number-interpretation)   evaluates to the number   limn→∞ (a1+ ··· +an)

f)   the infinite expression   a1+a2+a3+··· (sequence-interpretation)   evaluates to the sequence   (a1+ ··· +an)n≥1

g) Being p1, p2, p3, ··· successive primes,
      the infinite expression   p1-3+ p2-3 + p3-3+ ···   evaluates to the number   p1-3+ p2-3 + p3-3+ ···

h)   the infinite expression   9− 9^1+ 9− 9^2+ 9− 9^3+ ···   evaluates to the number   Σi =1 9− 9^í

According to me this is a quite peculiar way to use the verb 'to evaluate' (in the intro of the present text: "A series is thus evaluated by examining ...."); you can show sources?   I only saw it, meaning: given an expression (denoting a number), find the decimal representation of its value, exact or approximated. --Hesselp (talk) 21:37, 2 May 2017 (UTC)

You're apparently still not grasping the distinction between expressions and numbers. This fundamental impasse is your own, and no editor's responsibility, to correct. When you write "given an expression (denoting a number)" there is nothing wrong with that, but expressions that denote numbers are called numerals. Series are not numerals, but neither is the expression "1+1". So it seems like your confusion is actually unrelated to the subject of this article. If you want to discuss the general subject of the article, go to WP:REFDESK/MATH. You post one wall of text after another, and don't show any glimmer of understanding what others have said. I'm done engaging in this. Sławomir Biały (talk) 21:57, 2 May 2017 (UTC)
Unfortunately you made no judgments (true, false, ...) at the statements a - h. That makes it difficult, if not impossible, for me and others, to understand the ratio of your critisims.
You prefer 'numeral' over the longer 'an expression denoting a number'.  Okay, perfect.
But I don't grasp why you declare:   the expression "1+1" is NOT a numeral. (neither is the expression "1+1")
For in your post dated 12:46, 2 May 2017(UTC), you started with:
- the expression "1+1" evaluates to/has value the number  "2" .
Is there anyone who can explain why
- the expression  1+1  denotes the number  2,   and
- the expression  e+π  denotes the number  e+π ,
should not be correct as well? --Hesselp (talk) 20:00, 3 May 2017 (UTC)

An 'infinite expression' is an expression with infinite dimensions, or ... ?[edit]

The intro of the present text explains the meaning of "series" using:
The series of an given infinite sequence is the infinite expression that is obtained by placing terms side-by-side with pluses in between.
By 'infinite expression' is not meant an expression with infinite physical dimensions. Nor an expression of the type "1/0".
The Wikipedia article says: "an expression in which some operators take an infinite number of arguments".
That's sufficiently clear to most of our readers?  I doubt.
Moreover, that article has: "Examples of well-defined infinite expressions include infinite sums, whether expressed using summation notation or as an infinite series, ....". With a circulating reasoning, because 'infinite sum' is linked to the article named ....'Series (mathematics)'.   --Hesselp (talk) 21:50, 2 May 2017 (UTC)

Recent edits[edit]

I feel uncomfortable with recent edits by Sławomir Biały (April 28 to May 1) . These edits are aimed for improving the formal accuracy of the definition of series. They result in an article such that a reader (say, an engineer), who works everyday with series, may not understand this article, or, at least, could have the feeling that the series that are described here are not the same as the series that he uses to manipulate. In other words, the article is WP: too technical and biased. I have not reverted these edits, because the previous version suffers from the same drawbacks, and the effect of Sławomir's edits is simply to make them clearer.

What is wrong in the article: Firstly, it is wrong that a series is an expression. A series is a mathematical object which is commonly represented by an expression that allows manipulating it. It becomes clear that a series is not an expression, when one remarks that and are two different expressions for the same series. The second mistake is that series are presented as acually infinite sums, while, they are potentially infinite sums. Even formal series are generally defined by a process of completion, which is a formalisation of potentially infinite sums.

A third issue of the article is that the basic explanations are completely lacking. In fact, the concept of series originates in the counter-intuitive fact that adding, one after the other, infinitely many numbers (potential infinity), one may reach a finite result. This is Zeno's paradox of Achilles and the tortoise: Zeno divided the task (for Achilles) of reaching the tortoise into infinitely many subtasks, and deduced that Achilles can never reach the tortoise because he could not admit that the total time needed by this infinity of subtasks can be finite.

What to do? IMO, above example of Achilles and the tortoise should be summarized very early in the lead (first or second paragraph), a section "Motivation'" must be added (before "Definition") for explaining in details the conceptual difficulties with infinite sums, all technicalities must be thrown away from the lead, section "Definition" must be rewritten and split for moving the formal definition near the bottom of the article, ... I'll try to implement this, step by step, in the next days. D.Lazard (talk) 11:02, 4 May 2017 (UTC)

I agree with your overall plan. The perspective of actual versus potential infinity seems like a useful one to bring out, and I feel that Zeno's paradox is an excellent way to make that more concrete for the appropriate audience. Some summary content on completions of algebras could be added to the generalizations sections as well. Sławomir Biały (talk) 11:29, 4 May 2017 (UTC)

@D.Lazard.   What is meant:
a series is  a description  of the operation: adding one-by-one infinitely many quantities (line 1)
a series is  the operation : adding one-by-one infinitely many terms (line 16) ?
What a reader should think of: an operation that cannot be carried on (not 'effectively') ?
I'm curious to see how you define (based on reliable sources): "a convergent infinite adding operation",   "a alternating infinite adding operation"   "a geometric infinite adding operation"   "a Fourier infinite adding operation"   "the Cauchy product of two infinite adding operations"   "a power infinite adding operation"   and much more.
Please present a mature proposal for the intro-plus-definition part of the article. Here on Talk page, so not unnecessary disturbing our Wiki-readers . --Hesselp (talk) 17:30, 4 May 2017 (UTC)
I apologise for a typo: I have written "one by one" instead of "one after one". For your second remark: the square root is the operation consisting of computing the real number whose square is the input. This operation cannot be done effectively in the case of , as the result cannot be written down, being an infinite sequence of digits that cannot be written in a finite amount of time. The same occurs for series. For your last remark, nobody has ever used the formulation that you introduce, and I do not see why defining these strange and mathematically incorrect phrases. D.Lazard (talk) 20:47, 4 May 2017 (UTC)
@D.Lazard.   No, you don't have to apologize. You wrote "one after the other";  for me the "one-by-one" sounded just a little bit more familiar, and I didn't see any difference in content.
But: you don't answer my first question: is a series a 'description' or an 'operation'?
On your statement: "the square root operation is in many cases an operation that cannot be done effectively" (I'm inclined to say for short: an impossible operation, a void operation)  I plan to come back later. You are right of course when I interprete "operation", just as "calculation" and "evaluation", as: rewriting a number (or a function) given in the limit-of-the-sum-sequence-of-a-given-sequence-representation,  into the well known decimal representation.
About your last remark: please be concrete, and tell what formulations you see as 'never used', and what phrases used by me you see as strange and incorrect.   What's wrong and what's incorrect with:
"sequence  (a1+···+an) n ≥1  converges"     or    "sequence  a1,   a1+a2,   a1+a2+a3,  ···  converges" ,
and (better suited to verbal communication):   "sequence a1, a2, a3, et cetera  is (not) summable" ?
You've seen the number of hits by Google for <summable sequence> and <summable sequences> ? Quite remarkable is the much lower number of hits for <suite sommable> in French. -- Hesselp (talk) 22:57, 4 May 2017 (UTC)

@D.Lazard.   Some more remarks / questions.
A.   Footnote 5 in the current version of the article mentions Michael Spivak's book "Calculus" (1st edition 1967, latest(?) 2008). His chapter INFINITE SERIES starts with a box with:
        A sequence is summable if the sequence of its partial sums converges.
        In this case the limit of its partial sums is called the sum of the sequence.
Isn't this extremely close to the wording:
        A sequence with a converging sum sequence (= sequence of partial sums) is called summable.
        The finite limit is called sum of the sequence.
as used in this edit ?   If you know a more preferable alternative for the word 'summable', please show it.
B.   Your view on the 'mathematical object series' , I understand as being:  the operation (evaluation, calculation) producing (if possible) an expression denoting the decimal representation of the sum of a given sequence.
I'll incorporate this view in the text I plan to edit instead of the current one (recently judged as "too technical", "biased", "worth cleaning up", "rather of a mess").
C.   In this post you wrote:
      "It appears that this concept is not a simple one, as it involves the concept of infinity, which was not well
      understood nor well accepted before the end of the 19th century (this make your citation of Cauchy
      irrelevant for discussing the modern terminology; note that he avoided carefully to talk about infinity)."
I don't see your point with  'avoided carefully'.
For in Cauchy's "Cours d'Analyse" (1821) I read on page 4:   "Lorsque ...s'approchent indéfiniment ...est appellée la limite de ... ". (As...approaches infinitely ... is called the limit of ...).   And on the famous/notorious page 123:  "...une suite indéfinie...", "la somme s'approche indéfiniment...d'une certaine limite s", "n croît indéfiniment"   ("an infinite sequence", "the sum approaches indefinitely some limit s", "n increases indefinitely).
I don't see a substantial difference with the 'modern' view. Please elucidate why citing Cauchy as I did, is irrelevant? -- Hesselp (talk) 11:38, 5 May 2017 (UTC)

Professor Lazard's edit makes a good starting point for the article, and I think it is worth cleaning up to replace what is currently there. In any case, I think the lead can probably wait until last. The rest of the article is rather of a mess. Sławomir Biały (talk) 20:56, 4 May 2017 (UTC)
It strikes me that there is something about series, in the "potential infinity" sense, that is like lazy evaluation in computer science. "Potentially infinite" data structures like infinite series are realized as lazily evaluated (potentially) infinite sums, for example. That article also discusses potential infinities in an obviously relevant scope. See this, for example. If RSes can be found, it would be worth adding to the article. Sławomir Biały (talk) 23:46, 4 May 2017 (UTC)
This revert seems disruptive, since the editor has already been informed that no one is going to continue to engage with him here. Is there consensus to revert it? Sławomir Biały (talk) 13:12, 5 May 2017 (UTC)
I have long since given up any attempt at engaging Hesselp. You have my full support in attempting to improve this page, including whatever is needed to deal with this disruptive editor. --Bill Cherowitzo (talk) 15:01, 5 May 2017 (UTC)
I notice the same behavior as previously on the Dutch wikipedia, discussions with Hesselp on this topic turned out to be never-ending and non-convergent. Bob.v.R (talk) 00:55, 6 June 2017 (UTC)
Hesselp has been topic banned from this page (article and talk) for a duration of 6 months, starting 28 May 2017. Maybe a similar action could be done on Dutch WP. D.Lazard (talk) 15:58, 6 June 2017 (UTC)

Motivation for reposting an alternative intro+definition text,  6 May 2017[edit]

In Talk page, no user took part in discussion on the merits of the content of this text. So  'no consensus'  cannot be a valid reason to revert.

From the 'edit summary' 13:22, 5 May 2017:   "...most editors have already given up trying to communicate with you" .
That   "trying to communicate"   refers to reactions with no more relation to the content of the proposed text-section, than in phrases of the type:

- don't agree with proposed changes   - undocumented POV-pushing   - Hesslp doesn't understand what a series is   - this talk page is not for discussing personal opinions about the practice of mathematicians   - this is not mathematics, it is philosophy   - you have clearly a misconception of what is mathematics   - for being clearer, every line of Hesselp's post is either wrong, or does not belong to this talk page or both   - I reiterate my objection .

This reacting on attempts made to start discussion on the content, in this list:

-- Hesselp (talk) 10:20, 6 May 2017 (UTC)

Additional secondary source[edit]

To the list of  "Secondary sources supporting Hesselp's edits" (22:52, 27 April 2017, answering Wcherowi's remark 17:16, 25 April 2017  "...your edits are not supported by citations to reliable secondary sources...")  I add:
- R. Creighton Buck (1920-1998, University of Wisconsin),   Advanced Calculus, editions 1956, 1965, 1978 :   "An infinite series is often defined to be  'an expression of the form Σ1 an '.   It is recognised that this has many defects."
-- Hesselp (talk) 13:42, 8 May 2017 (UTC)

"Mathematicians agree on the concept of a series".  Is this true?[edit]

D.Lazard writes (15:14, 30 April 2017):   "Presently, mathematicians agree on the concept of a series, but as usual for concepts that have many applications, the formal rigorous definition is too technical for being understood by beginners, .....".
This 'agree on' seems to be not in accordance with the ongoing rewriting of the Definition section in the article. Not with the absence of a decisive unambiguous source. And not with the result of a survey, made around 2008. About eighty books on calculus were inspected, the results are shown below (press [show]). The original language was not always English; capital-sigma forms were seen as not different from  a1 + a2 + a3 + ··· .

This not very satisfactory situation, caused by the double meaning of 'convergence' in the 19th century, can be structured by accepting that:
- when 'series' is used denoting a mathematical object, it is synonym with 'sequence' (as in the 19th century and later), and
- in other cases 'series' is designating a certain kind/type of expression (or representation, or evaluation expansion, or maybe even more).
Instead of 'series expression' mostly the shorter 'series' is used.  But one has to realize that with 'convergent series' is not meant: 'the convergent mathematical object named series ', but: the convergent mathematical object denoted by the (type series) expression.
-- Hesselp (talk) 20:28, 8 May 2017 (UTC)

Like Carl below I don't want to get into a long discussion. But let me just take this opportunity to register my bafflement that you think an expression (even an infinite expression) cannot be a mathematical object. —David Eppstein (talk) 22:01, 8 May 2017 (UTC)
You can be right that I've said/written "an expression (even an infinite expression) cannot be a mathematical object". Please, specify in which post (so, in which context) I wrote this. (To be precise: you didn't say that I wrote this, but that you suppose that I think this.)
Let me say this on it. I'm quite convinced that, in order to have a good idea about what mathematics is and how it works, you should distinguish between the mathematical object 'itself', and the way it is (or: can be) expressed. (Expressed by written mathematical symbols, by written text, or verbally.) In this sense I see mathematical objects as different from expressions.   But, when 'expression' is seen as 'a string of discernible signs, you can study such strings extensively; so in this context such string-expressions can be called 'mathematical objects' with good rights. Is this what you meant with your remark?
You refered to "even an infinite expression".  In several attempts to define a concept 'series' I met this label 'infinite expression'.  But it remains unclear for me which condition should be fulfilled for an expression to be an infinite expression. Can you discern, infinite expression or not? :
a)         b)         c)         d)   .   -- Hesselp (talk) 09:38, 9 May 2017 (UTC)
None of those use infinitely many symbols. They are all finite expressions. However, some of them describe series (which if you like you can think of as infinite expressions); for instance (a) would usually be understood as referring to the series . This is not in principle different from the fact that as expressions and are different but that as numbers they are equal. —David Eppstein (talk) 15:43, 10 May 2017 (UTC)
I add three more expressions:   e)      f)       g)  and ask:
A.   Are the expressions labeled e, f, g  finite expressions as well?
B.   Which out of a - f are usually understood as referring to a series (describing a series)?
C.   Do you see "referring to a series" as meaning the same as "denoting a series" ?
D.   Please, show an example of an infinite expression.
E.   In the present text "series" is defined as being: an infinite expression (of a certain type). You write "If you like you can think of a series as being an infinite expression (an infinite tree labeled with symbols of various types)." That's not a definition as we are used to in mathematics, isn't it? -- Hesselp (talk) 18:49, 10 May 2017 (UTC)
Like other participants here, I don't see the point in continuing to participate in interactions with you. Your discussion here does not seem to be based on improving the article based on mainstream mathematical work, but about some sort of nitpicky definitional dispute which is mostly off-topic for this article. —David Eppstein (talk) 19:17, 10 May 2017 (UTC)
It was you, David Eppstein, who mentioned 'infinite expressions' (the central key-word in the Definition-section in the present text) in your first post (22:01, 8 May 2017), registering your bafflement about my ignorance.  Now, asked for an are away!
I repeat my questions A-E.   Somebody else?   Again, this is about the heart of the article - the definition. -- Hesselp (talk) 20:45, 10 May 2017 (UTC)
Your repeated faux-naive questions don't address my earlier comments. Instead, they make clear that you are not gaining any understanding out of this interaction, are not working towards clarification of the article, and are merely continuing to try to win points, trip up other editors, and push your idiosyncratic viewpoint. What is the point of playing that game? —David Eppstein (talk) 20:55, 10 May 2017 (UTC)
'don't address my earlier comments' ? See your posts 14:36 and 15:43, 10 May 2017.
And yes, I try to win the point that the present 'definition' is at best a self-referencing sentence. I proposed an alternative wording.  I don't see why that couldn't be seen as an attempt to improve/clarify the article. Questions A-E are still unanswered..... -- Hesselp (talk) 22:19, 10 May 2017 (UTC)

More precise terminology[edit]

In this edit Michael Spivak's Calculus (editions 1967, 1980, 1994, 2008) was added as reference, with as edit summary: "Citing a standard calculus text is sufficient to verify all content of these sections.".
Spivak advocates as  more precise  terminology over the  somewhat peculiar  standard language (using 'series'):
- summable sequence
- sum of a sequence
- absolutely summable sequence
- uniformly summable sequence
- Cesaro summable sequence
- Abel summable sequence
This terminology could help to clarify expressions with 'series' in the article as well. --Hesselp (talk) 21:37, 9 May 2017 (UTC)

A few comments[edit]

I have no desire to enter long discussions about this article, but I wanted to leave a few comments about this revision [6]:

  • Articles should be about mathematical objects, not directly about the words for them. So we avoid writing "The word 'group' is used to mean ..." or "The word 'series' is used to mean ..." whenever possible. Instead we write "A group is ..." or "A series is ...". There is another example of this at WP:ISATERMFOR. Similarly, the title (and section) "Situations in which the word 'series' is used" is too focused on the word series instead of the concept.
  • Remarks such as "No sources are found, presenting a non-contradictory description of such a mathematical notion, ..." come across as the opinion of an author rather than as encyclopedia-worthy knowledge. Our articles should not assert that all existing sources are contradictory. More likely, when someone claims that all existing sources are wrong, that person has misunderstood something or is promoting an unusual viewpoint.
  • The section "Definitions, common wordings" is not, in my opinion, written in ordinary mathematical prose. The spacing in "R e d u c t i o n o f s u m s a n d p r o d u c t s" is out of place and doesn't match any common style on Wikipedia. More generally, the style of the top few sections has too many odd spacings, too many lists and bullets, and does not read as ordinary prose. To the largest extent possible, Wikipedia articles should follow the conventions of all of other mathematical prose.

— Carl (CBM · talk) 15:44, 8 May 2017 (UTC)

@Carl. Thank you very much for your concrete comments.
On point 1:   I understand your remark. this case? Fortunately you add: "whenever possible".
Here we have a mathematical object: (in modern words) a mapping on N. The traditional word for what later on is normally named "sequence".  And we have a mathematical concept(?), a certain type of expression (a sign for the 'infinite summation function' plus a sign for a sequence as its argument). You may change the order of the two.
The same 'series-type' we meet when classifying representations (for numbers or functions), and when classifying expansions (for functions).
I'm afraid this cannot be combined in one phrase. I explained this in my article text.
On point 2:   I plan to smooth the content of this footnote. Maybe omit it completely. You are right that this sharp, maybe exaggerated wording is better suited for a discussion on Talk page.
On point 3:   On the unusual spacing in R e d u c t i o n o f . . . you're 100% right, I was lazy when I copied it from elsewhere. On the use of other extra spacings: you cannot see them as making the text, and the formulas, better readable? Enough to accept some deviation from standard style?
And on the use of more 'ordinary prose': maybe a question of taste as well. I shall reconsider this. I wouldn't take as an example the present text of the article. For me that's very far from any encyclopedic style. -- Hesselp (talk) 22:17, 8 May 2017 (UTC)
@Carl. Once more on your point 2.
Footnote 2, extending text-sentence 11 in [7], says:  "No sources ARE FOUND....OF SUCH a notion."   You refer to this words with ALL EXISTING SOURCES are contradictory/wrong. That's not the same. No reader can expect that an article is written by people who have studied ALL existing sources. I'll consider how to prevent misreading at this point. --Hesselp (talk) 16:17, 9 May 2017 (UTC)
Your excuse for the unusual spacing is troubling to me. You should not be copying text here from elsewhere. —David Eppstein (talk) 21:50, 9 May 2017 (UTC)
@David Eppstein. Don't be worried or troubled. I copied these lines from this post. The 'unusual' spacing I used the day before in this post as well, to make headings in a proposal for a longer edit (not meant as sector-headings in Talk page).
I understand that it is not easy to find the condition for an expression to be an infinite expression ? --Hesselp (talk) 08:44, 10 May 2017 (UTC)
How, not easy? It's just an infinite tree labeled with symbols of various types. —David Eppstein (talk) 14:36, 10 May 2017 (UTC)
David Eppstein, if you haven't seen this discussion you may want to read it. I don't believe it is worth responding in any way to Hesselp's numerous posts to this talk page unless they post something that looks like it might actually gain consensus. Mike Christie (talk - contribs - library) 15:22, 10 May 2017 (UTC)
I did see it, but thanks for the reminder. —David Eppstein (talk) 15:36, 10 May 2017 (UTC)

Equivalent definitions, again[edit]

See Wikipedia talk:WikiProject Mathematics#Equivalent definitions, again. Boris Tsirelson (talk) 16:47, 12 May 2017 (UTC)

Citations, observations, supposition[edit]

Attempting to find a way to some kind of consensus, I add the following lines to this Talk page.

Citations, taken out of longer posts on Wikipedia talk:WikiProject Mathematics
- Tsirel - 19:15, 12 May 2017: ".. in general an expression has no value (but in "good" cases it has);"   (Comment Hesselp: the dispute is about the question whether a series-type expression has (in "good" cases) a number as its value, or a series (For: "a series is denoted by an expression like ..+..+..+···"))
- CBM - 20:00, 12 May 2017: "... the definitions that are often given in the books lack something that would be present in a graduate level text."   (Comment Hesselp: No one has presented such a graduate level text in this Talk page.)
- CBM - 20:00, 12 May 2017: "...we should follow the sources and present the same general understanding that they convey.]   (Comment Hesselp: That's easier said than done, see survey in 09:38, 9 May 2017)
- CBM - 20:09, 12 May 2017: "If numerous sources all find it possible to discuss a concept without a formal definition, we can certainly do so as well."
- D.Lazard - 20:43, 12 May 2017: " In any case, a series is not a sequence nor a pair of sequences nor an expression. It is an object which is built from a sequence."   (Comment Hesselp: D.Lazard's edited since 09:50, 14 Februari 2017 seven times a version with:  "a series is an expression").
- Tsirel - 05:02, 13 May 2017: "What does it mean? A vague term whose meaning is determined implicitly by the context, case-by-case?"
- Taku - 23:10, 13 May 2017: "... a series is a more of a heuristic concept than an explicitly defined concept."

Observations   Studying the terminology used in the 19th (and a good part of the 20th) century, concerning the 'series-representation' of numbers (and of functions), we can see two noteworthy points.
(1) The word 'series' was used frequently in situations where we should use 'sequence' now. (Also German 'Reihe' in 'Folge'-situations, and French 'série' in 'suite'-situations.)   Cauchy introduces 'série' explicitely for a sequence with numbers as terms; much later Bourbaki seems to copy this by using 'series' for a sequence with terms allowing the existence of a 'sum series'. The names 'arithmetical series', 'harmonical series', 'Fibonacci series', etc. were in common use.
(2) The words converge/convergent/convergence were used in case the terms have a limit, as well as in case the partial sums have a limit. Cauchy seems to use the verb 'converger' for terms with a limit, and the adverb 'convergent' for partial sums with a limit; quite confusing.   And Gauss once remarks: (Werke Abt.I, Band X, S.400) "Die Convergenz einer Reihe an sich ist also wohl zu unterscheiden von der Convergenz ihrer Summirung ...." (The convergence of the sequence itself has to be distinguished from the convergence of its summation.)

Suppostion   This situation: two words (series and sequence) for one notion, and one word for two properties (limiting terms and limiting partial sums), caused ongoing confusion. More and more culminating in a belief in the existence of a third 'mathematical object', apart from 'sequence' and 'the sum sequence of a given sequence'. A mysterious object or notion, whose definition/description causes the difficulties mentioned in the citations above.
How about the idea of describing this historical roots of the present confusion, in the Wikipedia article?  Can this be seen as a description of the existing situation, or is this seen as OR ? -- Hesselp (talk) 20:20, 14 May 2017 (UTC)

I don't think the article should focus on the historical roots to any great extent, except perhaps in a section on history. Sources from the 19th century are not likely to be of much use in this kind of elementary article, and indeed there were many more terminological problems at that time (compare the common use of "infinitesimal" at that time). Every contemporary calculus book I have seen has the same concept of a series, although of course the wording may vary from one author to another. — Carl (CBM · talk) 01:21, 15 May 2017 (UTC)
I.   On the contemporary calculus books you have seen:
- You have seen Stewart ?   "If we try to add the terms of an infinite sequence, we get an expression of the form  ..+..+..+ ···  which is called an (infinite) series."   (Every time I try to add the terms of an infinite sequence I get - after some hours - a heavy headache, not a 'series')
- You have seen Spivak ? A sequence is called summable if its sum sequence converges. This terminology is usually replaced by less precise expressions.
- You have seen the 'Bourbaki'-followers: Buck, Gaughan, Maurin, Protter/Morrey, Zamansky, Encyclopaedia of Mathematics1992, Cauchy ?  A sequence with an existing sum sequence, is called a series.
The following wordings (taken from calculus books 1956 - 2008) are describing the SAME CONCEPT ?   How many readers of Wikipedia can 'see this cat' ?
- An (infinite) series IS   an expression of the form  ..+..+..+ ···
- An (infinite) series IS   a formal infinite sum.
- The formal expression  ..+..+..+ ··· IS CALLED an (infinite) series.
- An (infinite) series is   a sequence
- An (infinite) series is   a sequence  whose terms are to be added up.
- An (infinite) series is   the sum of the terms of a sequence.
- An (infinite) series is   an infinite addition of numbers.
- An (infinite) series is   a mathematical proces which calls for an infinite number of additions.
- An (infinite) series is   a sequence of numbers with plus signs between these numbers.
- We have an (infinite) series if, between each two terms of an infinite sequence, we insert a plus sign.
- An (infinite) series is   a sequence, followed by its sum sequence.
- An (infinite) series is what we get if we add all the terms of an infinite sequence.
- When we wish to find the sum of an infinite sequence we call it an (infinite) series
- The sum sequence of a given sequence is called an (infinite) series.
- The sum sequence of a given sequence is called the (infinite) series connected with the given sequence.
To CBM and others:   Present the mean value of LCM and GDC of this 15 wordings.
II.   Can you mention one or more titles (of calculus books you have seen) with a definition / description of "series",  NOT self-referring - explicitely or implicitely - with phrases like:
• a series is an expression of the form  ..+..+..+ ···,   combined with
• the expression  ..+..+..+ ···  refers to (denotes) a series. ? -- Hesselp (talk) 16:53, 15 May 2017 (UTC)
III.  @CBM:  In your edit summary Article 01:29, 15 May 2017 you emphasize:  ..the key definition up front, which needs to move directly to the SUM of a series.." .
Isn't that exactly the content of the fist few sentences of this edit ?   As that lines try to say:
The (series-type) expression       (with symbols for the summation-function, and for a sequence as its argument)
denotes / refers to       (in case of a valid - not a void - expression;  the "good" ones, Tsirel says)
the SUM number of the named sequence.       (or the SUM function in case of function terms)
(So now the expression ..+..+..+··· is not cycling back to "series" again.) -- Hesselp (talk) 17:43, 15 May 2017 (UTC)
Well, yes, I think there is a single concept of "series" that all these books are presenting. On the other hand, a long "wall of text" is not pleasant to read on a talk page, and I am not likely to continue reading them. Please take the time to express yourself succinctly. I don't find the version that describes "series" as an adjective to be particularly compelling. Actually, I don't see anything exceptionally faulty with the current phrasing of the article, which I have just read again. Of course everyone has their own way of saying things, I and I would write things differently if it was my article, but I don't see any deep issues at the moment. I do plan to do some more copyediting over time. — Carl (CBM · talk) 18:39, 15 May 2017 (UTC)
I, for one, am mostly disturbed by a single word in this article: the title "Definition" of Sect. 1.1. I understand that we have good reasons not to give a single (up to equivalence) mathematical definition. But in a mathematical article (even undergraduate) I would not call "definition" something that is not a mathematical definition. I'd better inform the reader shortly but honestly, why no definition. Such words as "definition", "theorem" and "proof" are somewhat sacred for me.Boris Tsirelson (talk) 19:05, 15 May 2017 (UTC)
I have similar misgivings about the word "Definition". Perhaps it is appropriate to point out that a series ("the sum of infinitely many terms") is a mathematical concept that does not have a generally agreed upon definition, just as "the area under a graph" is a concept that does not have a proper mathematical definition, but can be formalized in different ways depending on the circumstances. Sławomir Biały (talk) 20:25, 15 May 2017 (UTC)
Saying that there is "no" definition may be too strong, and some might even claim it would be "original research". Perhaps we could simply remove the subheading "Definition", or change it to a different word. — Carl (CBM · talk) 23:25, 15 May 2017 (UTC)
Carl, you are the first among us to know exactly the meaning of "definition", "theorem" and "proof" in mathematics. Boris Tsirelson (talk) 05:10, 16 May 2017 (UTC)

I agree that the header "Definition" has to be changed. IMO, this should accompanied by some restructuring of the article. I see a first section "Motivation" for regrouping the details of Achilles-and-the-tortoise paradox and other explanations (this would allows reducing the size of lead by replacing the corresponding paragraph of the lead by a single sentence), and a second section "Basic properties", which should be rewritten for avoiding too much repetitions of the content of the lead. D.Lazard (talk) 08:27, 16 May 2017 (UTC)

- Instead of the heading "Definition", I have in mind: "Names and notations".
- About recent changes in the text of the article:
• The self-referring "A series is an expression denoting a series" can't be found in the text any longer. Improvement.
• In the definition of 'series', the two-track construction "a series is an infinite sum, is an infinite expression of the form .." disappeared. Improvement.
• The "such as" regarding the capital-sigma notation. Improvement. (Maybe some more variants can be shown? As well as
  a1 + a2 + ... + an + ...  as variant of the pluses-bullets form.)
• The label "infinite expression" (instead of "expression") is still there. Although no criterion is found for decerning. See
   ,    ,      .
• The intro (almost at the end) says:  "When this limit exists, one says that the series is convergent or summable, and the limit is called the sum of the series.   And the present definition says:  "a series is an infinite sum,..". Combined we get wordings as:  "a summable infinite sum"  and  "the sum of an infinite sum".
I know there are books where you can find this; but it's not very nice and comprehensible.   Is it definitely OR to add that it's not unusual to say "summable sequence" and "sum of a sequence" as well?  I referred to Spivak (1956...2008) and many hits in Google.
- The third sentence in the present text says: "Series are used in most areas of mathematics,..".   Isn't it true that the content of this sentence can be worded as well by:  "Capital-sigma expressions and pluses-bullets expressions are used in most areas of mathematics".
Why are this notations so important? Because they express a method to denote/describe irrational numbers (and as an generalization also functions) by means of a regular-patterned sequence with more familiar rationals as terms (or 'easier' functions).
The usual word for such a method to describe mathematical objects by means of simpler objects, is "representation".   We have: the decimal representation, the continued fraction representation, the infinite product representation, and some more. Not the least important is, what could be called "the infinite sum representation" or - in honour of the famous term - "the series representation". The representation based on the summation function for infinite sequences.
So, instead of saying "series are important" (with the hard to define term 'series'), you could say "the series representation is important" (describable without mysterious words). Is this a so big change that you are going to react with: "impossible, clear OR" ?
- Last remark. Caused by personal circumstances I've to tell that I leave by now Wikipedia for at least a couple of weeks. I wish you fruitful discussions. Hessel Pot -- Hesselp (talk) 08:53, 16 May 2017 (UTC)

Formal sums[edit]

A recent edit by CBM brought my attention to the fact that our article says

While the most common uses of series refer to their sume, it is also possible to treat series as formal sums, meaning that no operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition.

However, elsewhere, we define "formal sums" as members of a free abelian group, and the phrase formal sum redirects to our article on free abelian groups. This differs in several ways from the meaning here, notably that the free abelian group members are sums of only finitely many nonzero elements and that their ordering is unimportant, neither of which is true for formal series. Would it maybe make sense to turn the formal sum into a disambiguation page that points to this article for formal series, as well as pointing to the other article for the other meaning? Are there other meanings than these two that should be considered? —David Eppstein (talk) 04:26, 15 May 2017 (UTC)

Series belong to a completion of the free abelian group. I think the article free abelian group could discuss this, and that may eliminate the need for a disambiguation. (Edit: Ordering is obviously a bigger problem, though.) Sławomir Biały (talk) 12:49, 15 May 2017 (UTC)
Do they really? The free abelian group on what generators, then? —David Eppstein (talk) 14:36, 15 May 2017 (UTC)
  1. A formal series is, normally, a formal sum whose terms are indexed by the additive monoid If the terms are numbers or belong to an abelian group, the standard notation becomes ambiguous (should the additions be done or not?). This is usually solved by remarking that the monoid is isomorphic with the multiplicative monoid This induces an isomorphism between formal series and formal power series. It follows that "formal series" is a phrase that is rarely used, except as an abbreviation for formal power series.
  2. A prototype for the free abelian groups of rank n consists of polynomials with integer coefficients, of degree less than n. As, when n increases the union of these abelian groups is the ring of polynomials, the completion suggested by Sławomir is simply the completion of the polynomial ring for the X-adic topology. That is, the ring of formal power series is the completion of the polynomial ring for this topology.
  3. My conclusion is that, in this article, we must avoid to talk about formal sums: With the definition of a series as an infinite succession of additions, talking of formal sums in section Definition is unnecessarily confusing, and, because of above remarks, section Formal series must be replaced by a section "Formal power series". D.Lazard (talk) 16:37, 15 May 2017 (UTC)
Formal series are not usually restricted to having integer coefficients, so the connection to free abelian groups seems specious to me. —David Eppstein (talk) 16:59, 15 May 2017 (UTC)
I think there may be some expositional advantages to merging the section on "formal series" (which I moved from the definition) into the section on power series. Of course not all power series are formal series, but I think the most common use of formal series is via formal power series. Moreover, the section on power series currently mentions generating functions, which is a key topic for formal power series. — Carl (CBM · talk) 17:04, 15 May 2017 (UTC)
To David Eppstein: You are right, one has to replace everywhere "free abelian group" by "free module" or "vector space". After all, a free abelian group is a free module over the integers. Thus one looses nothing by this replacement. Moreover, as presented in Formal sum, the phrase "formal sum" is confusing. In fact, as defined there, the abelian group of formal sums over a set S contains the formal differences of two elements of S. D.Lazard (talk) 17:36, 15 May 2017 (UTC)
To D. Lazard and/or David Eppstein: I think the point is that it's really the "free algebra", completed in a very strong topology. Sławomir Biały (talk) 02:11, 16 May 2017 (UTC)

IMO, this article should have a section "Formal power series" which is not a subsection. Beside the definition, this section should contain the fact that there is a bijection between sequences and formal power series; this bijection is used in many parts of mathematics and allows studying deep properties of integer sequences; examples are generating series, in combinatorics, and also Hilbert–Poincaré series in topology, and Hilbert series in algebraic geometry. This could also be useful for clarifying, for the reader, the relationship between sequences and series. D.Lazard (talk) 08:54, 16 May 2017 (UTC)

I have started this process. Please feel free to edit it more, keeping in mind that this should be a more elementary article when possible. I fear that some of the text may already be technical for a general article on series. — Carl (CBM · talk) 15:44, 22 May 2017 (UTC)