Talk:Series (mathematics)

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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 1,1,2,3,5,8 How to Sum this series
2 wrong definition
3 Formal definition?
4 My recent edit
5 Two notes
- 5.1 uncountable sums
- 5.2 Formula of F(x)
6 Illogical

[edit] 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)

[edit] 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):

— MFH: Talk 13:31, 26 Apr 2005 (UTC)

[edit] 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)

[edit] My recent edit

Okay, here's a breakdown:

There were two overlapping sections named "examples" and "types of series", or something like that. I merged them.
I moved the Taylor series for the exponential to the section on power series.
The section on convergence tests mentions absolute convergence, so I moved it to below the section on absolute convergence.
The tests were numbered; I changed them to an unnumbered list.
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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

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