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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 ${\displaystyle \sum _{n=1}^{b}ar^{n-1}.}$. 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 ${\displaystyle \sum _{n=1}^{\infty }ar^{n-1}.}$ is ${\displaystyle {\frac {a}{1-r}}}$  —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 ${\displaystyle \{a_{n}\}}$, define

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

Call ${\displaystyle S_{N}}$ the partial sum to N of the sequence ${\displaystyle \{a_{n}\}}$, 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
• Template: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

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

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

And from there is the list of references: ]

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)

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

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

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

mentioned in the article. Is it still true that calculating the sum is an open problem? 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. ${\displaystyle \sum _{n=1}^{N}n^{2}}$. —Kri (talk) 15:34, 19 October 2014 (UTC)

Yes, it can be referred to as a "summation index". Be care that in ${\displaystyle \sum _{n=1}^{N}n^{2}}$, 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 Misplaced Pages 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 ${\displaystyle \mathbb {R} ^{n}}$ 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 Misplaced Pages 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 ${\displaystyle a:\mathbb {N} \mapsto G}$ is a sequence denoted by ${\displaystyle a(n)=a_{n}}$.
I count three different notations for the same domain-${\displaystyle \mathbb {N} }$-function (sequence), four lines later a fourth version - ${\displaystyle (a_{n})}$ - is used.
Last remark: It's not correct to say that sequences (${\displaystyle (s_{n})}$ and ${\displaystyle (a_{n})}$, or simply ${\displaystyle s}$ and ${\displaystyle a}$) are subsets of semigroup ${\displaystyle G}$. -- 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 ${\displaystyle a}$ can be transformed into a limit number ${\displaystyle L}$.  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 ${\displaystyle a_{1}+a_{2}+a_{3}+\cdots }$  cannot be used in the generalized situation. And the other one, the capital-sigma form needs adaption (${\displaystyle i\in I}$ instead of ${\displaystyle i=1,2,3,\cdots }$ or ${\displaystyle i=1\ \infty }$ or ${\displaystyle i\geq 1}$ or ${\displaystyle i\geq 0}$).
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 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 ${\displaystyle a}$ is defined a
2. associated series Σ${\displaystyle a}$ (defined as: an ordered "element of the free abelian group with a given set as basis" - the link says).
3. To series Σ${\displaystyle a}$ is associated the
4. sequence ${\displaystyle s}$ of the partial sums of ${\displaystyle a}$ .
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'?  Misplaced Pages 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 ${\displaystyle a_{0}+a_{1}+a_{2}+\cdots }$ ,  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 ${\displaystyle a_{0}+a_{1}+a_{2}+\cdots }$ , 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:

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    ${\displaystyle a_{1}+a_{2}+a_{3}+\cdots }$    and    ${\displaystyle \sum a}$  or  ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$   expressing a number as the limit of the partial sums of sequence ${\displaystyle a}$, 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".
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 ${\displaystyle a}$ with terms ${\displaystyle a_{1},a_{2},a_{3}}$ et cetera (or starting with ${\displaystyle a_{0}}$) for which addition is defined, the sequence
${\displaystyle \quad a_{1},\quad a_{1}+a_{2},\quad a_{1}+a_{2}+a_{3},\ \ ...}$     is called  the sequence of partial sums of sequence ${\displaystyle a}$ .
Alternative notation:     ${\displaystyle (a_{1}+\cdots +a_{n})_{n=1,2,\cdots }}$  .
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   ${\displaystyle a_{1}+a_{2}+a_{3}+\cdots }$  (plusses-bullets form)   and   ${\displaystyle \sum _{n\geq 1}a_{n}}$  (capital-sigma form)
are sometimes used to denote the sequence of partial sums of sequence ${\displaystyle a}$ , 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 ${\displaystyle a}$ is a convergent series"  as well as  "series Σ ${\displaystyle a}$ converges"  are used to express that sequence ${\displaystyle a}$ is summable.   Similarly, "Σ ${\displaystyle a}$ is a divergent series"  and  "series Σ ${\displaystyle a}$ diverges"  are used to say that sequence ${\displaystyle a}$ 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 ${\displaystyle a}$  and  sequence ${\displaystyle a}$  are interchangeable in traditional clauses like:
- the sum of series Σ ${\displaystyle a}$,   the terms of series Σ ${\displaystyle a}$,   the (sequence of) partial sums of series Σ ${\displaystyle a}$,
  the Cauchy product of series Σ ${\displaystyle a}$ and series Σ ${\displaystyle b}$
- the series Σ ${\displaystyle a}$ is geometric, arithmetic, harmonic, alternating, non negative, increasing  (and more).

There is no standard interpretation for the limit of series Σ ${\displaystyle a}$.

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) ${\displaystyle x\rightarrow a_{n}(x-b)^{n},\ \ x\rightarrow a_{n}\sin ^{n}x+b_{n}\cos ^{n}x}$.
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. .

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 ]

-- 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 Misplaced Pages 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 Misplaced Pages. 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: ${\displaystyle a_{0}+a_{1}+a_{2}+\cdots ,}$ . . .
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 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 Misplaced Pages 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:

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

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

${\displaystyle \sum }$${\displaystyle _{i}\ a_{i}\ \times \quad \ c\quad \ \ \,=\,\sum }$${\displaystyle _{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)

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