Talk:Series (mathematics)/Archive 2

Meaning of divergence

Is a series which fails to converge necessarily called divergent?

e.g.

S = 1 - 1 + 1 - 1 + 1 - 1 +...

clearly does not converge, but it does not give an infinite value unless you rewrite it like

S = (1) + (1 + 1 - 1) + (1 + 1 + 1 - 1 - 1) + ...

= 1 + 1 + 1 + ...

Hmmm. OK, maybe there's some theorem that shows that any series which does not converge can be rewritten to give a larger value than any arbitrary value N, so i guess divergent makes sense. In any case, if that's the definition, then that's what wikipedia should present.

Boud 14:34, 7 Nov 2003 (UTC)

A series which does not converge is said to diverge. I think we tend to think of something which is divergent as somehow going to infinity but with seies this is not the case. See http://mathworld.wolfram.com/DivergentSeries.html for more details. -- Ams80 14:43, 7 Nov 2003 (UTC)

This is inded the case... just because it doesn't go off to infinity, it doesn't mean it's nto divergent. (It gets more confusing than this... infinite products that tend to zero are called "divergent"). Tompw 16:27, 7 October 2006 (UTC)

1,1,2,3,5,8 How to Sum this series

I just found this resource it's great!!

I have a series 1,1,2,3,5,8,13 Where first two terms are defined and therafter each term is the sum of the proceeding two. Some questions:

Does this kind of series have a name , does not seemto fit either geometric or arithmetic?

Is there a formula to get the Nth term? Is there a formula to calculate the sum of N terms?

That's the Fibonacci series. There is a formula for the Nth term, given on the page; don't know about sum of terms. Salsa Shark 12:46, 5 Jan 2004 (UTC)

That is not a series; it is a sequence. Michael Hardy 14:36, 5 Jan 2004 (UTC)

OK Michael, I'm still learning the terninology, is there a formula for the sum of N terms?.

Yes. I haven't worked out the details, but if you look at the article on Fibonacci numbers, you see that the sequence is the difference between two geometric sequences. Therefore you can apply the formula for a finite geometric series to each term separately. Michael Hardy 23:32, 5 Jan 2004 (UTC)

wrong definition

Is there a reference for the statement

a series is a sum ...

?

According to the definition I know,

• a series is the sequence of partial sums, (in this sense Fibonacci's numbers might even be a series... ;-)
• the sum of a series is the limit of this sequence, if it exists,
• a series is necessarily infinite, so the title of the first subsection is somehow meaningless.

I agree (well...) that a "handwaving" introduction is a good thing, but nonetheless it should be "unprecise" enough in order to be not wrong.

As to references of my understanding, a google search of "Definition series" gives on the first page the following links (besides many others with different meanings):

• http://planetmath.org/encyclopedia/Series.html
• http://faculty.washington.edu/edhong/previousquarters/320sp03/notes/tcss320A_14.ppt
• http://www.adeptscience.co.uk/products/mathsim/maple/powertools/calcII/html/L17-series2.html

— MFH: Talk 13:31, 26 Apr 2005 (UTC)

Formal definition?

I think a handwaving definition is not only acceptable, it's the only way to go. A series is a tool, just like a sum; it is safest not to attempt to define it as a mathematical object. We wish, for example, to write such things as

"$\sum a_{n}=A$ for some real number A"

and also

"$\sum a_{n}$ is a conditionally convergent series."

These statements are logically incompatible unless we agree that $\sum a_{n}$ is not a thing unto itself but a handy piece of notation. This is the view taken by Rudin, who writes in Principles of Mathematical Analysis:

"...or, more concisely,

(4) $\sum _{n=1}^{\infty }a_{n}.$

The symbol (4) we call an infinite series, or just a series... If {s_n} converges to s, we say that the series converges, and write

$\sum _{n=1}^{\infty }a_{n}=s.$

The number s is called the sum of the series...

He is careful to use the words "symbol" and "write". I think we ought to follow Rudin. MFH, your definition in the "Formal definition" section is internally consistent, but it isn't consistent with a statement of the form

$\sum a_{n}=A,$

since the left-hand side is a sequence and the right-hand side is not. Your definition would instead lead us to write

$\sum a_{n}\to A,$

which is a notation I have never seen before, and I'm sure you don't advocate it. Now, as I write this, I'm looking at the article itself, and it seems that we define neither of the above two kinds of "equations". Well, it's the first equation that is in common use, including elsewhere on Wikipedia, and an encyclopedic article ought to say what it means!

By admitting that the notation $\sum a_{n}$ is context-dependent, we can also sidestep a great deal of confusion. I freely admit that this goal is a major motivation for me.

So, who objects to me rewriting the relevant bits of the article with Rudin as a reference? Melchoir 02:19, 5 December 2005 (UTC)

I don't see the problem with the article the way it is. It states that the $\sum a_{n}$ notation is used to both denote the sequence of partial sums, and the actual limit of this sequence, if it is exists. Did I miss something? Oleg Alexandrov (talk) 02:55, 5 December 2005 (UTC)

I must be on crack. The text "Only in the latter case, i.e. if this sequence has a limit, the notation is also used to denote the limit of this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may omit the limits (atop and below the sum's symbol) in the former case." has always been there, yet I failed to read it. Wow.

Sorry.

Well, I'm still not quite humbled enough to let go. I don't think it's common to religiously omit the limits when one wishes to discuss the sequence as opposed to its limit. Shall we insert a sentence admitting that it's not always done, including in this article, and therefore some examination of context is still necessary? Melchoir 03:19, 5 December 2005 (UTC)

Well, from what I know the limits are never omitted to start with, either for the sequence of partial sums, or for its limit. Or not? Oleg Alexandrov (talk) 04:24, 5 December 2005 (UTC)

Um, I'm not sure what you're referring to by "to start with", but I like your edit to the article. Melchoir 05:11, 5 December 2005 (UTC)

Meaning, I think the only reason drop the limits from the series are laziness, that's how I see it. :) I never encountered series without limits. Oleg Alexandrov (talk) 05:42, 5 December 2005 (UTC)

My recent edit

Okay, here's a breakdown:

1. There were two overlapping sections named "examples" and "types of series", or something like that. I merged them.
2. I moved the Taylor series for the exponential to the section on power series.
3. The section on convergence tests mentions absolute convergence, so I moved it to below the section on absolute convergence.
4. The tests were numbered; I changed them to an unnumbered list.
5. I changed the phrase "the sum" to "a series" at the top of Absolute convergence.

I think that's all. Melchoir 02:21, 6 December 2005 (UTC)

Thanks, this is very helpful! I notice the item 3 a long while ago too, but did not bother to change it. :) Oleg Alexandrov (talk) 03:18, 6 December 2005 (UTC)

Two notes

First of all, from the article, "There is no serious definition for an infinite sum over an uncountable set." This is true on some level, but an integral is exactly an infinite sum over an uncountable set. I understand the difference between an integral and a sum, and what is being said in the article is essentially valid, but I think that sentence is a bit inaccurate, and maybe a little misleading. What I mean is, isn't a $\sum _{n=0}^{\infty }$ defined only for discrete n? The only logical extension of the sum to an uncountable range is the riemann integral, and that is certainly a serious definition... I don't know, it just seems misleading to say that there is no way to define a sum over an uncountable set. Also, I think the section on convergence tests should be expanded, at the very least to agree stylistically with the rest of the article, that is, using rendered math images instead of inline text. It would be a little easier to follow the definitions of the tests, and it would look nicer. --Monguin61 06:38, 12 December 2005 (UTC)

Note that an integral is not the same as a sum. In an integral you put more and more terms, but the weight of each term is decreased proportionally.

About series which contain uncountable number of terms. It is possible to define those. However, one can prove that if such an uncountable series is convergent, then it must have only a countable number of non-zero terms. So, as far as summation is concerned, countable is as high as you can go.

About using PNG images instead of inline text, that should be avoided per the math style manual. That is, one should use PNG images when one does not have a choice but not otherwise.

You are welcome to make changes to this article, as long as big changes are discussed in advance in here. Oleg Alexandrov (talk) 16:02, 12 December 2005 (UTC)

Why are there so many PNG images for formulas and expressions, here and elsewhere? It seems like the vast majority of them would work fine as inline text. The style manual, as far as I can tell, discourages inline images mostly, but also discourages images in general, but rendered images are used extensively throughout the math pages here. What I meant for this article was to change the convergence tests section from inline text, to images, but not inline images. In other words, similar formatting as the rest of the article. Wouldn't it be best to make the article consistent with itself by either changing the tests section to use images, or by changing the rest of the article to use inline text instead of images? I'm asking about this more to get a feel for the math style guidelines, I don't really care about this particular page so much. --Monguin61 21:04, 12 December 2005 (UTC)

Well, the math style manual calls for the use of inline png images to be minimized. I guess the people who wrote this article thought otherwise. So, if anybody is willing to help fix this article, that would be indeed encouraged. Oleg Alexandrov (talk) 21:56, 12 December 2005 (UTC)

I was under the impression that this: ∑b_n  is inline text, this: $\sum _{n=0}^{\infty }$ is an inline image, and this:

$\sum _{n=0}^{\infty }$

is not an inline image. Am I wrong? --Monguin61 22:04, 12 December 2005 (UTC)

That's right. Series better not be inline, as together with the limits of summation (which must be there) they take a lot of room. Oleg Alexandrov (talk) 22:24, 12 December 2005 (UTC)

Alright, thats what I thought. What I was pointing out is not that there are inline images in the article, because there arent many of those. What I wanted to discuss is that most of the math stuff is done using non-inline images, but there is one section which stands out stylistically, because it uses inline text. Any idea why that section was done like that? --Monguin61 22:31, 12 December 2005 (UTC)

Because many people worked on it, that's why. Just fix it, don't keep on asking. :) By the way, I would like to ask you to use an edit summary when you contribute, it helps others understand what your point is (like the "Subject" line in an email.) Thanks. Oleg Alexandrov (talk) 22:42, 12 December 2005 (UTC)

Oleg is right. An integral is not a sum. There is no way to understand an integral as a sum over an uncountable set. They are similar, however. I think the best way to understand their similarity is to realize that they are both integrals, one with the counting measure and one with the Lebesgue measure. The remarks in the article about how an uncountable sum must be nonzero countable many times proves that an uncountable sum cannot be regarded as an integral, at least not without heavy and qualitative modifications to the definition of sum. -lethe ^{talk +} 17:12, 30 March 2006 (UTC)

I think the notion that integration is a form of summation is pretty common, so I think some mention of the idea is appropriate in the article, even though it's a wrong idea. -lethe ^{talk +} 17:21, 30 March 2006 (UTC)

uncountable sums

It is indeed possible to have a sensible notoin of uncountable sums. I have written some text on the subject at User:Lethe/sum, and I intend to just drop it wholesale into the section about generalizations, but it occurs to me that this may be too long for that section. On the other hand, I don't really believe that uncountable sums are interesting enough to deserve their own article. I welcome your input. -lethe ^{talk +} 13:37, 30 March 2006 (UTC)

Maybe wholesale is too much. I'm currently editing it into a workable form. Stay tuned. -lethe ^{talk +} 15:14, 30 March 2006 (UTC)

I have finished my addition. There were some speculative things in my user subpage, but none of that into the article. Mostly, the wrong suggestion that there is no reasonable definition of uncountable sums was really bothering me, and I had to get rid of it. -lethe ^{talk +} 17:12, 30 March 2006 (UTC)

I didn't get the example of the sum of f(x) = 1 an uncountable number of times. What are you adding? Cardinals or ordinals? Albmont 17:30, 5 January 2007 (UTC)

Formula of F(x)

F(x)=1^n+2^n+3^n+x^n

when n=2, is there any formula for F(x)? when n=3, is there any formula for F(x)? Answers are in Talk:Euler-Maclaurin_formula#Some_Formula Jackzhp 03:55, 3 September 2006 (UTC)

See Faulhaber's formula. Michael Hardy 21:13, 3 November 2006 (UTC)

Illogical

"We say that this series converges towards S, or that its value is S" can something be done about this? Converging towards S and equalling S are two completely different identities. --JohnLattier 11:27, 2 November 2006 (UTC)

Note that the article does not say that the series equals S. On the other hand, the wording is unfamiliar to me, it's uncited, and it doesn't agree with the textbooks in arm's reach, so I will change it. Melchoir 15:42, 2 November 2006 (UTC)

What is equal to S is the limit of the sequence of partial sums. In some contexts it makes sense to identify that series with that. A relevant example involving finite sums, and no limits, is this: in some contexts, identifying "3 + 3" with the number 6 makes sense. In other contexts it does not; for example "3 + 3" is one partition of the number 6 and "4 + 1 + 1" is another. Michael Hardy 21:11, 3 November 2006 (UTC)

semi convergence

The section on semi-convergence does not include a definition (or a reference to one), and is therefore of very limited interest. Especially since I only opened the page to look up the definition...

How to find the formula for a Series?

I think there should be a guide to get the formula for a Series. For example, suppose I am solving a problem and I get 1 + 1/1! + 1/2! + 1/3! + ... There could be links to pages like "Series with General Term p(n) where p(n) is a polynomial", "Series with General Term p(n)/q(n)", etc. The e-series would be in "Series with General Term using n!" or something like that. Albmont 17:19, 5 January 2007 (UTC)

Recurring decimals?

Somewhere around the middle of the text, the text talks about recurring decimals converging to real numbers because of the completeness axiom. AFAIK, there is no such thing as a 'completeness axiom' - the real numbers are complete by construction (limits of Cauchy series blahblah). Also, a recurring decimal does not require completeness, since it is a rational number. I think that part might require a rewrite, or perhaps even left out (is it really relevant?) —The preceding unsigned comment was added by 134.58.253.130 (talk) 11:35, 29 January 2007 (UTC).

Completeness property of the set of real numbers

The completeness property is what distinguishes the reals from the rationals, and not only is it not an axiom (completeness has to be proven from the definition of real number), but there are two different types of completeness that are commonly discussed: Dedekind completeness and Cauchy completeness. These turn out to be equivalent in the reals, but are not equivalent in general. I have edited the page to say "completeness property" and point directly to Complete spaces. WikiWonki222 16:49, 30 January 2007 (UTC)

That's fine, but please recognize that yours is not the only way to deal with the real numbers. It is common to define the real numbers as a set of unknown composition satisfying certain axioms, one of which is completeness (in whichever sense). The fact that we can model this structure in set theory is reassuring, but that's all we're doing: constructing for ourselves a complete space, not investigating a space handed down by God and noticing that it happens to be complete. To say that completeness is somehow "not an axiom" misses the point. Melchoir 19:03, 30 January 2007 (UTC)

Sum Of Geometric Series

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 $\sum _{n=1}^{b}ar^{n-1}.$. 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 $\sum _{n=1}^{\infty }ar^{n-1}.$ is ${\frac {a}{1-r}}$  —Preceding unsigned comment added by Dandiggs (talk • contribs) 21:07, 31 January 2008 (UTC) 

Properties of Series

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 (talk • contribs) 10:05, 28 April 2008 (UTC)

Partial sum

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 $\{a_{n}\}$, define

$S_{N}=\sum _{n=0}^{N}a_{n}=a_{0}+a_{1}+a_{2}+\cdots +a_{N}.$

Call $S_{N}$ the partial sum to N of the sequence $\{a_{n}\}$, or partial sum of the series.

What more should one say? --Bdmy (talk) 21:36, 16 April 2009 (UTC)

Remainder

Remainder term redirects here but there is no introduction to the concept of remainder in infinite series on this page. --209.4.252.99 (talk) 19:24, 5 May 2009 (UTC)

Indian Mathematics

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.

Evidence

First, the source articles:

• Madhava_of_Sangamagrama
• Kerala school of astronomy and mathematics

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

• Leibniz formula for pi (redirect from Madhava-Leibniz)
• Integral test for convergence
• History of geometry
• History of calculus
• History of mathematics
• Numerical approximations of π
• Calculus
• Science in the Middle Ages
• Irrational number
• Timeline of mathematics
• Series (mathematics)
• History of astronomy (Possibly)
• Taylor series
• List of important publications in mathematics
• Mathematical analysis
• History of science
• James Gregory (mathematician) (how is Madhava even remotely relevant in this article?)
• History of trigonometry
• Mean_value_theorem

... 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)

Notation

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)

Definitions

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 213.80.200.218 (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

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
46.115.48.133 (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)

$-\sum _{n=1}^{\infty }{\frac {1}{10^{n}}}=-0.111\ldots =-1/9$

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?

I don't see the series

$\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}$

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"

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)

Terminology

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. $\sum _{n=1}^{N}n^{2}$. —Kri (talk) 15:34, 19 October 2014 (UTC)

Yes, it can be referred to as a "summation index". Be care that in $\sum _{n=1}^{N}n^{2}$, 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?

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 181.29.52.110 (talk) 00:17, 21 July 2015 (UTC)

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

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, ... (, a_i , ...)   is often replaced by the plus-signs form   a₁ + a₂ + a₃ + ... (+ a_i + ...)   or the sigma form   Σ_{i =1,2,...} a_i   .
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, ···, u_n, u_n+1, ···

be convergent, it is necessary and it suffices that increasing values of n make the sum

s_n = u₀ + u₁ + u₂ + ··· u_n-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 "...is 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 $\mathbb {R} ^{n}$ 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
Biały 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
Biały 13:31, 20 January 2016 (UTC)

How to denote a sequence?

To 166.216.158.233, 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 $a:\mathbb {N} \mapsto G$ is a sequence denoted by $a(n)=a_{n}$.
I count three different notations for the same domain-$\mathbb {N}$-function (sequence), four lines later a fourth version - $(a_{n})$ - is used.
Last remark: It's not correct to say that sequences ($(s_{n})$ and $(a_{n})$, or simply $s$ and $a$) are subsets of semigroup $G$. -- Hesselp (talk) 19:43, 10 April 2017 (UTC)

Index sets as generalization of N (subsection Definition)

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 $a$ can be transformed into a limit number $L$.  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 $a_{1}+a_{2}+a_{3}+\cdots$  cannot be used in the generalized situation. And the other one, the capital-sigma form needs adaption ($i\in I$ instead of $i=1,2,3,\cdots$ or $i=1\ \infty$ or $i\geq 1$ or $i\geq 0$).
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

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 $a$ is defined a
2. associated series Σ$a$ (defined as: an ordered "element of the free abelian group with a given set as basis" - the link says).
3. To series Σ$a$ is associated the
4. sequence $s$ of the partial sums of $a$ .
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

Line 3, quotation: "Summation notation....to 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 definition....is 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 $a_{0}+a_{1}+a_{2}+\cdots$ ,  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 $a_{0}+a_{1}+a_{2}+\cdots$ , 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

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:

::::::::::(press [show]-button →)

I n t r o d u c t i o n

In mathematics (calculus), the word series is primarily used for expressions of a certain kind, denoting numbers (or functions).
Symbolic forms like    $a_{1}+a_{2}+a_{3}+\cdots$    and    $\sum a$  or  $\sum _{n=1}^{\infty }a_{n}$   expressing a number as the limit of the partial sums of sequence $a$, are called series expression or shorter series.

Secondly, in a more abstract sense, series is used for a certain kind of representation (of a number or a function),  and also for a special type of such a series representation named series expansion (of a function, e.g. Maclaurin series, Fourier series).

And thirdly, series can be synonymous with sequence.  Cauchy defined the word series by "an infinite sequence of real numbers".[source: Cours d'Analyse, p.123, p.2, 1821, 2009]
This use of the word 'series' can be seen as somewhat outdated.

The study of series is a major part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics), through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

C o n t e n t s

D e f i n i t i o n s,   c o m m o n  w o r d i n g s
Given a infinite sequence $a$ with terms $a_{1},a_{2},a_{3}$ et cetera (or starting with $a_{0}$) for which addition is defined, the sequence
$\quad a_{1},\quad a_{1}+a_{2},\quad a_{1}+a_{2}+a_{3},\ \ ...$     is called  the sequence of partial sums of sequence $a$ .
Alternative notation:     $(a_{1}+\cdots +a_{n})_{n=1,2,\cdots }$  .
Example: The sequence 1, 2, 3, 4, ···  is the sequence of partial sums of sequence 1, 1, 1, 1, ··· ;  the sequence 1, 1, 1, 1,···  is the sequence of partial sums of sequence 1, 0, 0, 0,··· ;  this can be extended in both directions.

A series is a written expression using mathematical signs, consisting of
- an expression denoting the function that maps a given sequence on the limit of its sequence of partial sums
combined with
- an expression denoting an infinite sequence (with addition and distance defined).

Second meaning   The symbolic forms   $a_{1}+a_{2}+a_{3}+\cdots$  (plusses-bullets form)   and   $\sum _{n\geq 1}a_{n}$  (capital-sigma form)
are sometimes used to denote the sequence of partial sums of sequence $a$ , instead of the value of its eventual existing limit.

A sequence is called summable iff its sequence of partial sums converges (has a finite limit, named: sum of the sequence).

Convergent / divergent series   The combination convergent series shouldn't be interpreted literally, for an expression itself cannot be convergent or divergent.  By tradition  "Σ $a$ is a convergent series"  as well as  "series Σ $a$ converges"  are used to express that sequence $a$ is summable.   Similarly, "Σ $a$ is a divergent series"  and  "series Σ $a$ diverges"  are used to say that sequence $a$ is not summable.

Convergence test for series   Again, this traditional wording cannot be taken literally because 'series' is the name of an expression of a certain kind, not the name of a mathematical notion. An alternative is: summability test for sequences.

Absolute convergent series   This is the traditional naming for a sequence with summable absolute values of its terms. The alternative absolute summable sequence is not in common use.

Series Σ $a$  and  sequence $a$  are interchangeable in traditional clauses like:
- the sum of series Σ $a$,   the terms of series Σ $a$,   the (sequence of) partial sums of series Σ $a$,
  the Cauchy product of series Σ $a$ and series Σ $b$
- the series Σ $a$ is geometric, arithmetic, harmonic, alternating, non negative, increasing  (and more).

There is no standard interpretation for the limit of series Σ $a$.

S e r i e s  r e p r e s e n t a t i o n   o f   n u m b e r s   a n d   f u n c t i o n s
In some contexts the word 'series' shouldn't be seen as referring to a certain type of written symbolic expressions, but as referring to a special type of representation of numbers (and functions). Namely: defining a (irrational) number as the limit of the partial sums of a known infinite sequence of (rational or irrational) numbers. And in the case of functions: defining a function as the limit of the partial sums of an infinite sequence of functions (which are seen as 'easier' or more elementary in one way or another than the function represented by the limit).
Examples of the use of the word 'series' in this sense, can be seen in the final sentences of the introduction above, starting with "The study of series is a major part ...".

As comparable with the idea of series representation or infinite sum representation can be seen:  the continued fraction representation and the infinite product representation (for numbers and functions).

S e r i e s  e x p a n s i o n   o f   f u n c t i o n s
The combination 'series expansion' is used for a special type of series representation of functions. ('Series expansion of numbers '  is meaningless.)
A series expansion is a representation of a function by means of the infinite sum of a sequence of power functions of increasing degree, in one of its variables. Or functions like (for example) $x\rightarrow a_{n}(x-b)^{n},\ \ x\rightarrow a_{n}\sin ^{n}x+b_{n}\cos ^{n}x$.
The labels Maclaurin series, Taylor series, Fourier series shouldn't be seen as denoting expressions but rather representations of the type series expansion. So Maclaurin series should be understood as Maclaurin expansion, Fourier series as Fourier expansion, et cetera. [Source: WolframMathWorld series expansion and Maclaurin series].

P o w e r  s e r i e s
"Power series" can be used
- as synonym for "Maclaurin expansion", and
- for a series expression which includes a sequence of power functions with increasing degree.

C a u c h y   a s   s o u r c e   o f   c o n f u s i o n
Cauchy, in his 'Cours d'Analyse' (1821) made an important, but quite subtile, distinction between the meaning of 'to converge' and 'being convergent':
- a sequence (French: suite) can converge (both French and English) to a limit, versus
- an infinite sequence of real numbers (named 'série' by Cauchy) having its sequence of partial sums converging to a limit, the first sequence named 'une série convergente ' .
Only a tiny difference between 'sequence' and 'series', but an essential one between 'converging' and 'convergent'.
This imprudent choise caused permanent confusion around the use of the word 'series'(e.g. in the German translations of 'Cours d'Analyse' of 1828 and 1885), until the present day.

[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: $a_{0}+a_{1}+a_{2}+\cdots ,$ . . .
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 (a_n), (s_n)".  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

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:

$\sum$$_{i}\ a_{i}\ +\,\sum$$_{i}\ b_{i}\ =\,\sum$$_{i}\ (a_{i}+b_{i})$

$\sum$$_{i}\ a_{i}\ \times \,\sum$$_{i}\ b_{i}\ =\,\sum$$_{i}\ (a_{1}b_{i}+\cdots +a_{i}b_{1})$     (sequence $(|a_{i}|)$ or sequence $(|b_{i}|)$ summable)

$\sum$$_{i}\ a_{i}\ \times \quad \ c\quad \ \ \,=\,\sum$$_{i}\ (a_{i}\,c)$ .
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)"

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

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^∞ u_n   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  {a_n}  is, or is not, summable is conventionally replaced by the statement that the series   Σ_{n =1}^∞ a_n   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

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

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

::::::::::this edit in Talk  (to open: press the [show]-button)

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)

-- 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 $a_{n}$ 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 $a_{n}$ 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+···+a_n) _{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

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 $a_{n}$ 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 $1+1+1+\cdots$, 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    $a_{1}+a_{2}+a_{3}+\cdots$    and    $\sum a$  or  $\sum _{n=1}^{\infty }a_{n}$   expressing a number as the limit of the
        partial sums of sequence $a$, are called series expression.  'Series expression' is often shortened to just 'series'.
I use the short notations $a$ for a mapping on N (a sequence) and $\sum a$ as alternative for $\sum _{n=1}^{\infty }a_{n}$ (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 "$\lim _{h\to 0}\sin h/h$" 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 ... ?

@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   Σ_{i =1}^∞ a_i   evaluates to the infinite expression   a₁+a₂+a₃+···

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

f)   the infinite expression   a₁+a₂+a₃+··· (sequence-interpretation)   evaluates to the sequence   (a₁+ ··· +a_n)_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   Σ_{i =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 ... ?

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

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 $a_{0}+a_{1}+\ldots +a_{n}+\ldots$ and $\textstyle \sum _{n=0}^{\infty }$ 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)
or
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 ${\sqrt {2}}$, 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+···+a_n) _{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)