# Talk:Series (mathematics)

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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 (talkcontribs) 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:

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

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

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

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

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

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

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

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

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

## 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? [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. ${\displaystyle \sum _{n=1}^{N}n^{2}}$. —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, ... (, ai , ...)   is often replaced by the plus-signs form   a1 + a2 + a3 + ... (+ ai + ...)   or the sigma form   Σi =1,2,... ai   .
Two remarks:
1. The plus-signs form and the sigma form are also used for the sum of a (and sometimes as well as for the sequence of partial sums of a).
2. In almost all modern texts the words convergent/divergent/to converge/to diverge, in combination with the word 'sequence', apply to the terms, and not to the partial sums.   In some older texts (mostly 19th century, following Cauchy) the verbs are used only in combination with 'sequence', and the adjectives only with 'series'; the word 'convergence' doesn't occur. See Bradley R.E., Sandifer C.E., 2009, Cauchy's Cours d'analyse - An Annotated Translation

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

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

I agree that "series" and "sequence" are fundamentally the same concept. However, we need to remember that articles like this are supposed to talk to as general an audience as possible and not just to mathematicians. I don't think these ideas will improve the article, especially not in the lead. McKay (talk) 02:42, 20 January 2016 (UTC)
"The same concept". Okay. So why should we go on with a Wikipedia article strongly suggesting (lying?) that 'series' and 'sequence' stand for different mathematical things? Cannot we find simple words to say that in certain situations 'sequence' is frequently replaced by 'series' (and in that case: 'summable' by 'convergent', and the comma notation by the plus-signs or the sigma notation)?
The present text starts with "This article is about infinite sums." Is it clear for a general audience what is meant with "sums that aren't normal sums"? --Hesselp (talk) 16:14, 20 January 2016 (UTC)
Firstly the sentence "This article is about infinite sums" is not a part of the article, it belongs to a disambiguation hat note.
"The same concept". No. Although in common language "series" and "sequence" are almost synonymous, in mathematics, they refer to concepts that are different although strongly related (to each series one may associate the sequence of its partial sums, as well as the sequence of its terms, and to each sequence one may associate the series whose terms are the differences of successive elements). This is the reason for which I have moved "In mathematics" in the article. To see that series and sequences are different concepts, it suffices to consider the product: The product of two sequences is obtained by multiplying together the terms that have the same index. On the other hand, the product of two series is a series that has a completely different definition; it is chosen in order that, if the series are (absolutely) convergent, the sum of the series product is the product of the sum of the series factors. D.Lazard (talk) 18:23, 20 January 2016 (UTC)
@D.Lazard. 1. The very first sentence "...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.) 13:34, 21 January 2016 (UTC)