Talk:Divergent series

1 − 1 + 1 − 1 + · · ·
''For example, Cesàro summation assigns the divergent series


 * $$1 - 1 + 1 - 1 + \cdots$$

the value $$1 \over 2$$.''

Well, no, the average of the sum of N terms &rarr; 0, evidently. But perhaps some other example was meant. Such as


 * 1 + 0 + 1 + 0 + ....

Charles Matthews 16:20, 7 Sep 2004 (UTC)


 * Not so: the average of PARTIAL sums is important, no? and the average of 1, 0, 1, 0,  etc. is? —The preceding unsigned comment was added by Danwbartlett (talk • contribs) 00:00, 11 March 2005  (UTC)


 * Charles, It depends what summation method you're using.
 * S = (1-1)+(1-1)+(1-1)... gives us S=0, but..
 * S = 1+(-1+1)+(-1+1)+(-1+1).... gives us S=1, and then again..
 * S = 1-(1-1+1-1+1-1...) gives us S=1-S, and thus 2S=1, and therefore S=1/2.
 * so I guess the third example must known as Cesaro Summation as described above. —The preceding unsigned comment was added by 203.59.206.158 (talk • contribs) 09:33, 28 January 2007 (UTC)


 * For sums of this series, please refer to Summation of Grandi's series or related articles. Melchoir 19:30, 29 January 2007 (UTC)

Removal
Cut this out, for a couple of reasons.

"== Proving an infinite series is divergent ==

There is a quick test to prove an infinite series is convergent or divergent.

Given an infinite series, if

$$\lim_{n \to \infty} \sum_{n=1}^\infty a_k \ne 0$$

then $$ a_k$$ is said to be divergent.

Example
Prove whether the following is convergent or divergent.

$$\lim_{n \to \infty} \sum_{n=1}^\infty cos \frac{1}{k^2}$$

Solution:

$$ \lim_{n \to \infty} cos \frac{1}{k^2} \to 1$$

$$1 \ne 0$$

Therefore, this series is divergent."

Reason (1): the test is mis-stated. A necessary condition for convergence is that the ak &rarr; 0. Reason (2) is that this really belongs on the basic infinite series page. Charles Matthews 06:19, 12 Feb 2005 (UTC)

This is the first of the convergence criteria given at infinite series, in fact. Charles Matthews 06:21, 12 Feb 2005 (UTC)


 * Oops, I looked there. I need to look more carefully. Tygar 06:49, Feb 12, 2005 (UTC)

Second sentence?
The second sentence of this article doesn't seem to add anything. (No one is reading this in order to learn what "antonym" means.) But I wanted to see if there are objections before deleting it. (I would probably put the link to "convergent series" on the word "converge" at the end of the first sentence.) Dchudz 23:26, 6 November 2006 (UTC)

issue with the Hahn-Banach paragraph
"The operator giving the sum of a convergent sequence is linear, and it follows from the Hahn-Banach theorem that it may be extended to a summation method summing any bounded sequence."

We talk about summing series, and not sequences, right? And isn't it the sequence of partial sums that needs to be bounded, and not terms of the series? If there aren't objections, I'll change it to:

"The operator giving the limit of a convergent sequence is linear, and it follows from the Hahn-Banach theorem that it may be extended to a summation method summing any series whose sequence of partial sums is bounded."

(Or anyone who is confident that I'm right could just go ahead and make the change.)

Dchudz 23:34, 6 November 2006 (UTC)


 * I think it's okay to speak about summing a sequence. However, a convergent sequence is something else as a convergent series, so you're right that it should be changed. I replaced it by "The operator giving the sum of a convergent series is linear, and it follows from the Hahn-Banach theorem that it may be extended to a summation method summing any series with bounded partial sums." which is a slight reformulation of what you proposed. -- Jitse Niesen (talk) 01:07, 7 November 2006 (UTC)


 * Yeah, that's good. Thanks. 130.58.219.209 02:54, 7 November 2006 (UTC)

Is there a way to do the Hahn-Banach extension so that your summation method ends up having "stability" (regularity and linearity are clear...)?Dchudz 18:20, 14 December 2006 (UTC)

Definition of "converge"
The second sentence states, "if a series converges, the individual terms of the series must approach zero," but this contradicts the definition given on the infinite series page, which states, "this limit can have a finite value; if it does, the series is said to converge..."

Obviously, there are finite values other than zero. Am I missing something? Phlake 10:29, 12 November 2006 (UTC)


 * The first sentence you quote says:
 * "if the series $$ a_1 + a_2 + a_3 + \cdots $$ converges, then $$ \lim_{i\to\infty} a_i = 0.$$"
 * The second sentence you quote says:
 * "the series $$ a_1 + a_2 + a_3 + \cdots $$ converges if the limit $$ \lim_{n\to\infty} \sum_{i=1}^n a_i $$ has a finite value."
 * The first case is about the limit of the individual terms. The second case is about the limit of partial sums. I hope that answers your question. -- Jitse Niesen (talk) 11:07, 12 November 2006 (UTC)


 * Aha. That's what I missed.  Thanks.  Phlake 13:40, 19 November 2006 (UTC)

Stability vs. translativity
Shouldn't the stability condition be that A(s) = A(s′ )+s0? Or is stability different from, and incompatible with, translativity? --mglg(talk) 01:06, 15 April 2007 (UTC)


 * Yes, I think you are right - the stability condition should be A(s) = A(s′) + s0. I have fixed this in the article. Gandalf61 08:26, 15 April 2007 (UTC)


 * No, I don't think so. When we talk about the sequence, we are not talking about the terms in the series; we are talking about the sequence of partial sums of the series.  So e.g. the series 1 + 1/2 + 1/4 + ... is associated with the sequence (1, 3/2, 7/4, ...).  If we truncate this sequence: (3/2, 7/4, ...), the associated sum is 3/2 + 1/4 + ...  Therefore, stability "should" mean that the truncation of the sequence (of partial sums) is sent to the same value by A as the sequence itself.  I'm changing this back in the article, but if you disagree with me we can talk about this further. Kier07 09:11, 15 April 2007 (UTC)


 * I attempted to clarify this in the section, and also modified the definition of linearity. Linearity should be on all defined sequences not just the convergent ones, otherwise it is a weaker property than regularity. --Ørjan 11:28, 15 April 2007 (UTC)


 * Ah, yes - my bad. It is clearer now. Gandalf61 12:42, 15 April 2007 (UTC)


 * Question... in the linearity part, when we say that A is linear if it is a linear functional on the sequences on which it is defined, are we saying that if A is defined on r and s, then it is defined on r + s, and A(r + s) = A(r) + A(s)? Can we make this really explicit in the article?  Thanks... Kier07 18:31, 15 April 2007 (UTC)


 * Yes, that is the way I would usually interpret something being a linear functional. The domain of definition is a vector space.  --Ørjan 20:25, 15 April 2007 (UTC)


 * Ouch, we got burnt over at 1-2+3-4 because of this, and it seems that neither phrasing can be right, in terms of sequences. (I assume Melchoir referring to Hardy there has it right, although I hope someone here with access to the book will actually look it up.)  Unfortunately phrasing the precise condition in terms of sequences is not very elegant (unlike for series, where you just take off the first term and move it out of the A as above), but I will attempt to do it and then I'll leave it up to you to decide if perhaps using the series terms is better. --Ørjan 21:15, 15 April 2007 (UTC)


 * Sorry for stepping in the same tarpit; I should have checked this talk when I first noticed the reversion. The question of whether stability should be defined on {sn+1} or {sn+1-a0} is moot if we assume translatability (which is a weak form of linearity).  But I don't know which is the essential stability, nor whether anyone really makes use of stability in the absence of translatability. But I did notice that the article 1 + 2 + 4 + 8 + · · · refers to totally regular summation method, which redirects here.  Any idea what that property is? –Dan Hoey 22:57, 15 April 2007 (UTC)


 * I don't have Hardy in front of me right now, but I recall that total regularity means that the method also must agree with ordinary summation where the ordinary sum is infinity. Melchoir 18:03, 16 April 2007 (UTC)

I think this section is confusing or incomplete. I think it could use some further references if possible. I was unable to find stability or translativity in Hardy, or via google searches, for example. Are translativity and stability different or not? (I think both terms should be referenced, since it will aid those looking for further information.) The info at Michon's site is clearer ( I added reference, but may have done it incorrectly). I would like a reference that explains further the absence for Borel Summation. Asllearner (talk) 17:57, 3 July 2013 (UTC)

"Totally regular"
Totally regular summation method (seen in 1 + 2 + 4 + 8 + · · ·) redirects here, but the article does not say what "totally regular" means as opposed to just "regular". At least I can't tell what it means. Does any one else know? -- 136.159.61.31 16:56, 17 September 2007 (UTC)


 * If I recall correctly, when a series diverges to infinity, a totally regular summation method also sums the series to infinity. A merely regular summation method only has to agree with convergent summation when the sum is finite. But I may be forgetting some subtleties; you should consult Hardy for the real definition. Melchoir (talk) 08:42, 5 May 2008 (UTC)
 * (Hmm, looks like I said the same thing in the above section!) Melchoir (talk) 09:00, 5 May 2008 (UTC)

Abel Summation
There is no unique page for Abel Summation. When searching for "Abel Sum," you are redirected to this page, which gives (in my opinion) a pretty terrible explanation of Abel sums. However, these pages give a much better explanation of the topic. I think it should be given its own page. -- BlueRaja (talk) 04:07, 5 May 2008 (UTC)


 * I agree, although it might be a little tricky to figure out the proper relationship between the new article, the section in this article, and Abel's theorem. Melchoir (talk) 08:51, 5 May 2008 (UTC)

Norlund Means
In the section on Norlund means there is an error.

The combinatoric symbol (n + k - 1 / k - 1) should be equal to Gamma(n + k) / (Gamma(n + 1) * Gamma(k)). The term Gamma(n + 1) in the denominator has been omitted.

This would also be equal to 1 / ((n + k) * Beta(n + 1, k)) if you want to include that as well.

131.193.140.154 (talk) 17:03, 5 August 2009 (UTC)

Fixed. 92.225.64.186 (talk) 06:22, 4 January 2010 (UTC)

"Summable"
I notice that this article is a redirect from "Summable". Would it be possible to include a reference to summable in the text of the article?

I'm interested in summable because I've seen it in Henri Cartan's book Elementary Theory of Analytic Functions of One or Several Complex Variables, &copy; 1995, Dover. A family of formal series is said to be summable if, for any integer k the order of each formal series in the family is &ge;k for all but a finite number of indices.

Confluente (talk) 04:54, 27 Oct 2011 (UTC)

Convergence of summation methods of divergent series
Can we say that applying two different summation methods to one divergent series, we will get one result?

Is there proof of the Legendre rule that an error in the evaluation of the asymptotic series due to its truncation at any one term is of the order of the first term discarded? Taulalai (talk) 20:53, 13 October 2012 (UTC)

Alternative summation methods for the harmonic series?
This page points out that for several notable divergent series, for whom no traditionally-defined sum can exist, mathematicians have come up with 'thinking-outside-the-box' sums calculated through various creative alternative methods; and the wikipedia explains several of these in detail on their respective pages (with articles on series such as 1 − 1 + 1 − 1 + · · ·, 1 + 2 + 4 + 8 + · · ·, 1 + 1 + 1 + 1 + · · ·, 1 + 2 + 3 + 4 + · · ·, and so forth; and articles on techniques such as Cesàro summation, Ramanujan summation, and so forth.)

But what about the harmonic series? Nowhere on the Wikipedia can I find either a proposed 'creative' summation of it, nor an explanation for the lack thereof. It seems like the obvious question to ask. Doops | talk 04:44, 15 January 2014 (UTC)

Remarks about Euler in section "history"
I don't know what the rationale of the remarks about Euler and his "not first defining what a divergent sum is" and the problems/contradictions in the 18'th and 19'th is. I remember as snippet from an Euler-translation, where he discussed the problems of the use of divergent series with some pros and contras and trying to clear up various uses by Bernoulli and Leibniz. He then stated just that what the WP-article claims to be missing: "that the divergent series should have just that value which it had in a mathematical operation where this divergent series results from". He then illustrated this by the arithmetic operation 1/(1+x) and the algebraic operation of developing this into a power series in x by "long division", and that the series which appears if we insert a value into x, that series should have the same value as the fraction 1/(1+x). I don't remember that perfectly, but I think it was not explictely expressed, that the same should be applicable for 1/(1+x)^n in the same manner, even if the expression by a power series resulting by the formal long division would display a divergent series for some actual values of x (but it follows immediately from Euler's proposal to use that value "whenever there is an algebraic expression from where the divergent series stems".   I don't remember the source, maybe it was in some of Ed Sandifer's "How Euler did it" (now at ams.org with the shortcut "HEDI") but it should be existent in the original latin version in the Euler-archives which are online.

Gotti 09:15, 11 August 2014 (UTC)

Added: I've found a reference in K.Knopp's work on infinite series in Chap 13 he cites a letter of Euler to Goldbach:(7 VIII 1745) "(...)so I've given this new definition for the sum of each series: Summa cuiusque seriei est valor expressionis illius finitae, ex cuius evolutione illa series oritur." (the german part in Knopp's book translateb by me into english) — Preceding unsigned comment added by Druseltal2005 (talk • contribs) 09:43, 11 August 2014 (UTC)

Gotti 09:17, 11 August 2014 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

--Gotti 09:44, 11 August 2014 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Added link to simple example on Wikiversity.
Hi. I added a link to this simple calculus-based problem on Wikiveristy at the top of the article. Usually such interlinks go to the bottom of the article, but I thought this example might be of interest to readers before they read your article.

Say the word and I will move the interlink to the bottom, or even remove it entirely if you wish.--guyvan52 (talk) 16:40, 6 September 2014 (UTC)

A more general definition
A summation is a partial function $$\Sigma$$ on the set of sequences $$(a_n)$$ into an additive group such that (i) for all $$k$$ $$\Sigma(a_n)=a_0+\cdots +a_k + \Sigma(a'_n)$$, with $$a'_n = 0$$ for $$n\leq k$$ and $$a'_n=a_n$$ for $$n>k$$ and (ii) $$\Sigma(0)=0$$. — Preceding unsigned comment added by 2804:14C:8780:970C:1D16:289B:465E:B6CC (talk) 02:32, 17 June 2018 (UTC)

Using Grandi's series as an example of a divergent series
In the spirit of BOLD, revert, discuss: I added:


 * It is not necessary for a divergent series to approach infinity, only that it not have a finite limit. For example, Grandi's divergent series,
 * $$1 - 1 + 1 - 1 + \cdots,$$
 * does not approach any limit at all, finite or infinite. Its partial sum is either 0 or 1, depending on whether it stops with an even number of terms or an odd. Thus, a "divergent series" need not "diverge" in the everyday sense of the word.

JayBeeEll reverted it (among other changes) saying "this introduced errors and inappropriate editorializing; the purpose of the lead section is to summarize the article)".

Grandi's series was already in the lead section, along with another example which has an infinite limit, so it seems it is not objectionable to have examples in the lead section. I think it would be useful to point out to readers that the definition implies a divergent series might not have any limit at all, not just an infinite limit.

Is it an error to say that the partial sum is either 0 or 1? True, if you rearrange the sequence of terms, you can get lots of other partial sums, as the article on Grandi's series explains. But I think one would call the rearrangement a different series, as it is a different process of summing terms. If not, that is an error that can be fixed.

I don't think it's editorializing to say that alternating between 0 and 1 is not "diverging" in the everyday sense of the word. I could have cited to a dictionary, but this is common knowledge, isn't it? — Preceding unsigned comment added by Editeur24 (talk • contribs) 22:53, 18 December 2020 (UTC)

Deleting mention of Oresme and harmonic series in the lead section
In the spirit of BOLD, revert, discuss discussion:

I deleted:


 * The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.

JayBeeEll reverted it (among other changes) saying "this introduced errors and inappropriate editorializing; the purpose of the lead section is to summarize the article)". He probably didn't mean this deletion in particular, since it doesn't seem to match his criticisms.  I thought it was too tangential to be in an article on Divergent series, especially in the lead section.

Shall I delete it again? — Preceding unsigned comment added by Editeur24 (talk • contribs) 16:51, 19 December 2020 (UTC)

Someone Should Add Examples
This article would benefit hugely from examples. How about doing the 1-1+1-1+... series for each summation method? And maybe another example-- I don't know the topic, but there must be classic examples to illustrate the difference between the definitions. If we want to create an article useful to people who don't already know the contents, examples should be a high priority.editeur24 (talk) 16:52, 25 December 2022 (UTC)