Talk:1 + 2 + 3 + 4 + ⋯

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Vote for featured article The infinite sum stepping stair postulate with solution of -1/12 is so amazing after performing a smoothed curve regularization. I would like to suggest this article be featured on front page Wikipedia.

Merge[edit]

Why don't we merge the mathematically interesting results of this series with their respective articles and then just use this page as a redirect or disambig.--Cronholm144 22:22, 12 May 2007 (UTC)

Why would we do that? Melchoir 09:13, 13 May 2007 (UTC)

well because this article as it stands now doesn't really meet the standards for article inclusion. I know you are attached to divergent series, but every one doesn't deserve an article--Cronholm144 09:16, 13 May 2007 (UTC)

What standards would those be? Melchoir 09:51, 13 May 2007 (UTC)
Cronholm144, I have a couple of suggestions: (i) it would help if you were more specific about which article or articles you are proposing to merge this page into - it is difficult to respond usefully to a very generic proposal; (ii) if you wanted a wider response to your various merge proposals around divergent series, you could outline your arguments at Wikipedia talk:WikiProject Mathematics. Gandalf61 10:05, 13 May 2007 (UTC)

I am aware of this, all of my comments were made as I was working in a different vein, I did not realize that there were dedicated editors behind all of these articles. I have left my last word on the topic on Melchoir's talk page.--Cronholm144 10:10, 13 May 2007 (UTC)

Medium term organisation[edit]

Although 1-2+3-4+... generated some controversy, I was happy that it was featured, as I am a big fan, and have already expressed that in its maths rating. Cronholm's recent reaction to these articles on divergent series was rather hasty, but despite this, I do think there is case to be made for some reorganisation. Although Wikipedia is a work in progress, it is also a work in use, and Cronholm's experience may reflect a more general reader's reaction. I do not think these articles are in conflict with WP:NOT (an indiscriminate collection of information) but other readers and editors may not agree. I therefore wonder if there is not something to be gained by pooling some of the resources in these articles, at least while they undergo further development.

As far as I can tell, these articles can be discriminated in two ways. First, there is the contrast between divergent geometric series and divergent series with zeta and eta function regularization. Second, there is the contrast between series with alternating signs, and those without. At the very least, this suggests to me that this article could usefully be merged with 1 + 1 + 1 + 1 + · · · which at present adds little further information other than an anecdote.

Similarly, I see no reason why the divergent geometric series examples could not be merged, with the exception, of course, of Grandi's series, which deserves its own article (and also the 1 + 1 + 1 + 1 + · · · anecdote is more appropriately covered in the zeta-function context). Such a merger could take place either within the article itself, or in a separate article on examples of divergent geometric series. I look forward to hearing your views. Geometry guy 23:51, 13 May 2007 (UTC)

I can certainly see 1 + 2 + 3 + 4 + · · · and 1 + 1 + 1 + 1 + · · · being merged, as it is extremely hard to find a source that says anything about one of those series without naturally applying to the other. Even primary sources like Euler and Ramanujan seem to lump them together.
I could also understand 1 + 2 + 4 + 8 + · · · being merged into Divergent geometric series, because there is almost no mathematics that applies to 1 + 2 + 4 + 8 + · · · but not equally well to other geometric series. Most of that can be found at Two's complement anyway. However, you'd really have a devil of a time trying to shoehorn 1 − 2 + 4 − 8 + · · · in as well. That series has a special history; it can be Euler- and Borel-summed, which doesn't work for positive series; and the particular ease of Euler summation is unique. Melchoir 02:21, 14 May 2007 (UTC)

Thanks for this helpful and expert answer! This is worth thinking more about, I believe, but for me it will have to be tomorrow. Geometry guy 02:38, 14 May 2007 (UTC)

Don't get it...[edit]

Isn't 1 + 2 + 3 + 4 + · · · equal to the Cauchy product of 1 + 1 + 1 + 1 + · · · with 1 + 1 + 1 + 1 + · · · ? If so, 1 + 2 + 3 + 4 + · · · = -12·-12 = 14. I must be missing something... JocK 20:55, 2 July 2007 (UTC)

I think this just shows that methods of summing divergent series do not in general commute with the Cauchy product. So sum of Cauchy product \ne product of sums. Gandalf61 08:39, 3 July 2007 (UTC)

On an even more basic note, the article begins without any context. Someone with a basic knowledge of high school mathematics will likely understand most of the terms and symbols in the first section, but will be utterly bewildered by the statement that a sum of positive numbers yields a negative result. A sentence or two addressing this confusion would be very useful and important. See the article "Zeta Function Regularization", which appropriately places 'sum' in quotation marks. — Preceding unsigned comment added by 128.138.141.214 (talk) 21:20, 4 May 2012 (UTC)

Proof by Induction[edit]

One of the proofs I like, is the proof by induction showing (1/2)n(n+1) + n+1 is equal to (1/2)(n+1)(n+2). Hope that makes sense.--138.253.162.174 (talk) 17:18, 26 November 2008 (UTC)

Well duh, (1/2)n(n+1) + (n+1) is equal to ((1/2)n+1)(n+1), which is equal to (1/2)(n+2)(n+1). Besides, what does this has to do with 1 + 2 + 3 + 4 + ... 24.1.201.172 (talk) 01:20, 22 May 2010 (UTC)

Inadequate explanation[edit]

The statement in the lead that 1+2+3+4+... sums to a definite result and that this sum of positive terms is a negative number (-1/12) is not just "interesting". It is spectacularly and wonderfully counter-intuitive to readers not already versed in the Riemann zeta function (i.e. to virtually all Wikipedia readers). This would be a fascinating article if it delivered a good explanation of this result. Why doesn't it? By contrast, the article on 1-2+3-4+5+... does an excellent job (illustrations and all) of presenting a counter-intuitive result in a manner that requires nothing more of the reader than some basic junior high school level algebra.

In the already crowded field of Wikipedia math articles that have great potential but blow it on opaque text that is incomprehensible to readers without a university math degree, this one must surely be in the top 10. --Ross Fraser (talk) 00:00, 20 July 2012 (UTC)

An easy to understand explanation[edit]

If someone wanted, they could transcribe the proof presented here: https://www.youtube.com/watch?v=w-I6XTVZXww&feature=youtu.be

Basically, if

S_1=1-1+1-1+1-1+...=1/2,
then
S_2=1-2+3-4+5-6+...=1/4
and
if S=1+2+3+4+5+6+...
then
S-S_2=4+8+12+...=4*S
and so
3S=-S_2=-1/4
and S=-1/12
— Preceding unsigned comment added by 85.76.117.165 (talk) 14:14, 10 January 2014 (UTC)

to clarify. already the first step S_1=1-1+1-1+1-1+...=1/2 is wrong. the sequence of partial sums of this series alternates between 0 and 1. so the series of partial sums is (1,0,1,0,1,0, ..) and it doesn't converge (doesn't have a limit). then if you ASSUMED (the WRONG thing) that S_1 actually exists/converges, the next steps would be correct. however, no result follows from this, in the same way that you can conclude anything you want from the false equation "1 = 0". it troubles me that some guy does those videos on youtube, and has obviously no knowledge of the most basic mathematics (university level this is week 3 of a 5 years), but expresses those false things with the authority of some postdoc and doesn't scrutinize his own results, when the FIRST line of this article says the series is DIVERGENT. then in the end "yeah this is an odd result, but then i guess many physics results are counterintuitive, so this can, after all, be correct". sorry, this is no justification, this is not a mathematical approach and no physicist will say that this sum is really -1/12 (i am one). the series is divergent and will remain divergent until the end of time, the -1/12 is not totally incorrect but one has to be very careful in the wording, what's certainly false is that "the series of natural numbers converges to -1/12".
please be responsible and do correct mathematics when you do youtube videos or if you sum in a "non-standard way" then you need to explain this "generalized meaning" of "summation". (simmilarly when we talk of functions with lebesgue integrals, when we are really talking about equivalence classes of functions. we just say functions but it's not correct, we have to be more specific outside of this context.)

92.196.5.246 (talk) 14:07, 17 January 2014 (UTC)

easy to understand is good, but easy doesn't mean correct. and if it's wrong, there's nothing to understand, because when there's no logic behind the result what's there to understand? the calculation is kind of random. the limit of 1 - 1 + 1 - 1 + .. is taking to have an arbitrary value (1/2). any other value could have been chosen with the same reasoning (why the arithmetic average not the geometric average which would be 0) and the result would have been different. 92.196.5.246 (talk) 14:10, 17 January 2014 (UTC)
That's too pessimistic. The intuitive manipulations may appear random at first, but they aren't. It should be possible to relate the intuitive manipulations to manipulations of formal Dirichlet series that converge on some common open set, ultimately mirroring a proof that zeta(-1)=-1/12 by comparing with the Dirichlet series of the eta function, where eta(0)=1/2 is not at all arbitrary. Melchoir (talk) 20:46, 17 January 2014 (UTC)
This is whats done in the response to comments on the numberphile video[1]. Three complex series are defied
S_1(z)=1+\sum_{n=1}^\infty (-1)^n \left[(n+1)^{-z}-n^{-z} \right]
S_2(z)=\sum_{n=1}^\infty (-1)^{n+1} n^{-z}
S(z)=\sum_{n=1}^\infty  n^{-z}
Substituting z=-1 gives the series S_1, S_2 and S above. It can be shown
S(z)-S_2(z)=2^{1-z}S(z) putting z=-1 gives S-S_2=4 S and ultimately S=-1/12.--Salix alba (talk): 07:58, 27 January 2014 (UTC)

I've gone ahead and added a slightly different version of the simple explanation, including the crucial step of multiplying by 4. This argument was given by Ramanujan, and that alone should justify repeating it, if only for historical commentary! Melchoir (talk) 05:49, 27 January 2014 (UTC)

January 2014 traffic[edit]

The traffic for this month is shown at http://stats.grok.se/en/201401/1_+_2_+_3_+_4_+_⋯. Starting from January 9, traffic is 4000 views/day, which I assume is due to Numberphile's video at https://www.youtube.com/watch?v=w-I6XTVZXww. Then, January 17 caught an additional 10,000 views. Does anyone know what caused the latter spike? Melchoir (talk) 03:18, 18 January 2014 (UTC)

I think is just the spread of the numberphile video, being mentioned a lot. EG BoingBoing, slate. --Salix alba (talk): 06:52, 18 January 2014 (UTC)
Good to know, thanks! Melchoir (talk) 01:42, 28 January 2014 (UTC)

Did Euler sum this series?[edit]

There are plenty of authors who claim that Leonhard Euler wrote that 1 + 2 + 3 + 4 + ⋯ = −1/12, including some of the links in the "External links" section. And yeah, okay... it does sound like something Euler might have written. However, the claims are always vague about when and where Euler wrote it down, and I can't find the equation in Euler's works!

Does anyone know if Euler really did write down the equation? Or is it nothing more than a rumor? Melchoir (talk) 01:42, 28 January 2014 (UTC)

Explaining revert[edit]

To explain this revert [2], I removed the following prose:

This is not actually a summation, but merely a number describing the series, which indicates that the series is divergent. It is comparable to a derivative, which indicates the slope (not the value) of a line. In the famous formula seen below, the = sign refers to a number which describes the sum, and does not mean "equals" in the traditional sense
  1. There's no distinction between the result of a generalized summation method and "merely a number describing the series". Every kind of summation, including the usual definition, is just a number that describes a series. Of course, we do need to distinguish between the generalized sum of −1/12 and the usual definition of summation for convergent series. But I think it's enough to say (in both of the lead paragraphs) that the series diverges and to put "sum" in quotation marks.
  2. The comparison with a derivative is a very interesting idea, since regularizing a sum can be thought of as picking out the constant part of an expansion in some parameter, just as the derivative picks out the linear part of the Taylor expansion. However, the analogy is also misleading, since a function with a derivative also has a value; the present series has a regularized sum but not a convergent sum.
  3. The "=" sign does mean "equals" as long as the "+ · · ·" on the left-hand side of the equation is interpreted using one of the summation methods in question. In that context, the equation states that the generalized sum is precisely equal to −1/12. Anyway, the cited sources don't say anything about the equals sign.

I'm not saying that the lead is perfect as it stands, either! Clarifications can be made as necessary, and we can subtly tailor the wording to avoid giving calculus students the wrong idea. But we should make sure that we don't hang so many scary caveats on the result that it's drained of all meaning. Melchoir (talk) 06:29, 28 January 2014 (UTC)

-1/12 is NOT the sum of that series. you need to explain a little more clearly why you are putting "sum" inside parantheses. If you don't like my explanation, feel free to EDIT what I wrote... but you shouldn't just delete it. The version you are reverting to provides no explanation whatsoever for why "sum" is not the correct word to use, and its causing confusion leading people to believe the sum of all positive integers is a negative number. that is NOT correct. clearly this article needs more caveats. — Preceding unsigned comment added by 68.185.193.1 (talkcontribs) 09:24, 28 January 2014‎

I think you're exaggerating the situation. The version I reverted to does provide an explanation for why "sum" is not the correct word to use: Because the series 1 + 2 + 3 + 4 + · · · diverges, it does not have a sum in the usual sense of the word. It would be reasonable to say that the explanation is insufficient, but it's disingenuous to say that there's no explanation whatsoever!
The usage of "sum" in the context of generalized summation methods is well-known and can be found throughout Hardy's monograph. For that matter, the word is used in Ramanujan's second letter, which is quoted in the body of the article. All that said, I can see how it could cause confusion in the lead section, so I'll replace it with more specific claims about the particular summation methods used.
In general, my objection about the caveats is that they shouldn't be attached to every single statement. It's just tedious. I'll try restructuring the lead to combine the caveats in the first paragraph. I think this is a reasonable compromise: it should satisfy the need to make the point as forcefully as possible, without spreading FUD to the rest of the content. Melchoir (talk) 20:04, 28 January 2014 (UTC)

Summability > Zeta function regularization > The Last Equation (added within 24 hrs.)[edit]

The equation-in-question is as the following:

-3\zeta(-1)=\eta(-1)=\lim_{x\to 1^-}\left(1-2x+3x^2-4x^3+\cdots\right)=\lim_{x\to 1^-}\frac{1}{(1+x)^2}=\frac14

Dirichlet Eta function is not continuous at s = -1, therefore it would be incorrect to say:

  • \eta(-1)=\lim_{x\to 1^-}\eta(x)
  • \eta(-1)=\lim_{x\to 1}\eta(x)
  • \eta(-1)=\lim_{x\to 1^+}\eta(x)

In fact, the last one would be all right. Anyway, the first one wouldn't be all right; hence, it is incorrect to say:

\eta(-1)=\lim_{x\to 1^-}\left(1-2x+3x^2-4x^3+\cdots\right)

You could say that "my analytically continued eta evaluates into 1/4 at -1" if you want, but that would not be equal to the sum 1-2+3-4+\cdots. It would at best be equal to the limit \lim_{x\to 1^-}\left(1-2x+3x^2-4x^3+\cdots\right). All in all, that particular equation is actually:

-3\zeta(-1)=\eta(-1)\ne\lim_{x\to 1^-}\left(1-2x+3x^2-4x^3+\cdots\right)=\lim_{x\to 1^-}\frac{1}{(1+x)^2}=\frac14

Thanks for reading. — Preceding unsigned comment added by ThoAppelsin (talkcontribs) 04:32, 1 February 2014 (UTC)



Well, there are a couple things going on here. First, the eta function is continuous at s = -1, so it would be correct to say \eta(-1)=\lim_{x\to 1}\eta(x). (Edit February 1: I meant, \eta(-1)=\lim_{x\to- 1}\eta(x)) However, that's aside from the point. The limit \lim_{x\to 1^-}\left(1-2x+3x^2-4x^3+\cdots\right) is the Abel sum of the (divergent) numerical series 1-2+3-4+\cdots, which is the (formal) value of the Dirichlet series 1^{-s}-2^{-s}+3^{-s}-4^{-s}+\cdots at s = -1. Since the eta function is equal to the Abel sum of that series everywhere, evaluating the limit tells us eta(-1).
Since it's been challenged, I'll go ahead and find a source for this... Melchoir (talk) 04:43, 1 February 2014 (UTC)
...Here's the diff with the references: [3]. Melchoir (talk) 06:35, 1 February 2014 (UTC)
By the way, about continuity at -1, you might have been thinking of the Dedekind eta function. I know I've made that mistake more than once! Melchoir (talk) 08:15, 1 February 2014 (UTC)




\eta(-1)=\left(1-2+3-4+\cdots\right)
Is that right? If so;
\eta(-1)=\lim_{x\to 1^-}\left(1-2x+3x^2-4x^3+\cdots\right)
implies:
\left(1-2+3-4+\cdots\right)=\lim_{x\to 1^-}\left(1-2x+3x^2-4x^3+\cdots\right)
which is not true. The 1 − 2x + 3xx − 4xxx + ⋅⋅⋅ is not continuous at x = −1. Dirichlet Eta may or may not be, but the series 1 − 2x + 3xx − 4xxx + ⋅⋅⋅ isn't. — manual signature: comment added by ThoAppelsin (talkcontribs) 15:08, 1 February 2014 (UTC)


Those equations are all true if (1 − 2 + 3 − 4 + · · ·) is interpreted as an Abel sum. But be careful with the notation: the limit \lim_{x\to 1^-} is the limit as x approaches 1 from below, NOT the limit as x approaches −1. I'll make that clearer in the article.
The series 1 − 2x + 3xx − 4xxx + ⋅⋅⋅ is convergent for −1 < x < 1. On that interval, the sum of the series is 1/(1 + x)2. The limit as x approaches −1 does not exist. But the limit as x approaches 1 does exist, and it equals −1/4. By definition, then, this is the Abel sum of the series 1 − 2 + 3 − 4 + · · ·. Melchoir (talk) 20:25, 1 February 2014 (UTC)


What does "the limit as x approaches 1 from below" mean? My guess is that it means "the limit as x approaches 1 from left or from lower values", could also be below I guess, if we tilt our heads 90° clockwise. If it does not mean anything else than my guess, then the notation was all clear from my point. Actually, I have to say that it used to be clear with that good old notation, and not so much anymore; it is the first time I am seeing the notation that is currently being used.
I concur with the first three sentences. However, as it is in the third sentence, in the fourth sentence, expression you are referring to is unclear to me. With "But the limit as x approaches 1 does exist, and it equals −1/4."; if you referring to the expression 1(1 + x, then it is all right; if you are, however, referring to the expression 1 − 2x + 3xx − 4xxx + ⋅⋅⋅, then sentence #4 is incorrect. The reason why I can concur with the third sentence despite being uncertain about its reference, is that it is true for both of the expressions. — manual signature: comment added by ThoAppelsin (talkcontribs) 23:10, 1 February 2014 (UTC)


It's a well-known problem that a string of written symbols can represent either a formal series or the sum of that series. The meaning is determined by context. In this case, when I equate the limit of an expression with a number, we can conclude that the expression must represent a function R -> R. This function is the sum 1 − 2x + 3xx − 4xxx + ⋅⋅⋅, AND it equals 1(1 + x for all x in an open set of the form (1-epsilon, 1).
Anyway, I think we're straying from discussing the article. It currently states:
-3\zeta(-1)=\eta(-1)=\lim_{x\nearrow 1}\left(1-2x+3x^2-4x^3+\cdots\right)=\lim_{x\nearrow 1}\frac{1}{(1+x)^2}=\frac14
Do you have a suggestion relating to those equations? Melchoir (talk) 23:36, 1 February 2014 (UTC)


I do:
-3\zeta(-1)=\eta(-1)={1}-{2}+{3}-{4}+\cdots\ne\lim_{x\nearrow 1}\left(1-2x+3x^2-4x^3+\cdots\right)=\lim_{x\nearrow 1}\frac{1}{(1+x)^2}=\frac14
— manual signature: comment added by ThoAppelsin (talkcontribs) 23:53, 1 February 2014 (UTC)


Are you claiming that \eta(-1)={1}-{2}+{3}-{4}+\cdots? Melchoir (talk) 00:04, 2 February 2014 (UTC)


I am. — manual signature: comment added by ThoAppelsin (talkcontribs) 00:40, 2 February 2014 (UTC)


Well, there's the problem. How do you define the right-hand side? Melchoir (talk) 00:46, 2 February 2014 (UTC)


Addition of the numbers that follow the rule n⋅(−1)n+1, starting with n = 1, increasing n by one for each subsequent number, doing this till forever. The variable a in the following would be it, though I don't think int would be able to contain that. Maybe a long long int... nah, that probably wouldn't make it either...
int main ( ) {
  int a = 0, b = 0, c = -1;
  while( 1 ) {
    b ++;         /* increase b by 1     */
    c *= -1;      /* multiply c with -1  */
    a += b * c;   /* increase a by b * c */
  }
}
— manual signature: comment added by ThoAppelsin (talkcontribs) 01:12, 2 February 2014 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── That's not helpful. Let me just compare the two kinds of summation that are involved here. To heighten the contrast, I'll write u() for the usual sum of a series and A() for the Abel sum of a series. Then for Re(s) > 0, we have

\eta(s)=u(1^{-s}-2^{-s}+3^{-s}-4^{-s}+\cdots).

However,

\eta(-1)\neq u(1-2+3-4+\cdots)

because the right side is undefined, and

u(1-2+3-4+\cdots)\neq\lim_{x\nearrow 1}u\left(1-2x+3x^2-4x^3+\cdots\right)

because the left side is undefined. If we were so inclined, we could write these two non-equations on the same line:

\eta(-1)\neq u(1-2+3-4+\cdots)\neq\lim_{x\nearrow 1}u\left(1-2x+3x^2-4x^3+\cdots\right)

But that doesn't really get us anywhere. On the other hand, switching to Abel summation, for all s, we have (see Knopp p.491)

\eta(s)=A(1^{-s}-2^{-s}+3^{-s}-4^{-s}+\cdots)

and by definition of the Abel sum,

A(1^{-s}-2^{-s}+3^{-s}-4^{-s}+\cdots)=\lim_{x\nearrow 1}u\left(1^{-s}-2^{-s}x+3^{-s}x^2-4^{-s}x^3+\cdots\right).

Putting the previous two equations together,

\eta(s)=\lim_{x\nearrow 1}u\left(1^{-s}-2^{-s}x+3^{-s}x^2-4^{-s}x^3+\cdots\right).

In particular, by taking s = −1, we get

\eta(-1)=A(1-2+3-4+\cdots).

Meanwhile, by definition of the Abel sum,

A(1-2+3-4+\cdots)=\lim_{x\nearrow 1}u\left(1-2x+3x^2-4x^3+\cdots\right).

Putting the previous two equations together,

\eta(-1)=\lim_{x\nearrow 1}u\left(1-2x+3x^2-4x^3+\cdots\right).

This amount of repetition and extra notation would be way too much to add to the present article. I write it here to clarify the above discussion. Melchoir (talk) 02:55, 2 February 2014 (UTC)


Until the part "But that doesn't really get us anywhere." I am all good, but then this explanation takes one detour from the right, leaving me on the main track because I am not familiar with the waypoints it's trying to give me. I have and already actually had looked up for the "Abel Sum", and the best I've got is Abel's summation formula, which I cannot relate to this. Then I checked the Knopp p.491, which (unless I am looking at something different here) only says that this is equal to this, and that is equal to that. There also is a disclaimer on top that it won't deal much with proofs or anything. Maybe it tells more than this, but it fails to deliver those points if they exist, or at least it fails to deliver those points to me.
As said, it doesn't get us anywhere, and it shouldn't really, because it doesn't go anywhere in particular itself. What is this "Abel Summation"? If it is not a summation in the world of mathematics where 2 + 2 = 4, why are you ascribing and why shall I ascribe any importance to it? Sincerely speaking, shouldn't simply any calculation, summation method, what-so-ever be dismissed as soon as it claims that adding up 1, 2, 3 and so on would end up with something, when applied with regards to all its guidelines?
I guess this is the end, I don't think I will be getting a reasonable, but truly reasonable, explanation to what Abel Summation is or how this switching can happen. The detour will be a mystery for me, I won't deal with it and take it as a truth, but I can take it as a belief that many Doctors of Philosophy choose to believe in, and will try to respect their beliefs. — manual signature: comment added by ThoAppelsin (talkcontribs) 03:57, 2 February 2014 (UTC)
There's a short descripiton at Divergent series#Abel summation, which Abel summation currently redirects to. It should probably be a separate article, but I guess no one's gotten around to writing it. (You could try requesting it at Wikipedia:Requested articles/Mathematics#Complex analysis.) About switching between methods in general, there are various abelian and tauberian theorems that relate different summability methods. In the case of the Dirichlet eta function, the important result is kind-of sort-of in the spirit of an abelian theorem: Abel summation of its Dirichlet series gives you an analytic function on the whole complex plane, which implies that it's consistent with any other method of analytic continuation: it truly is the Dirichlet eta function. This is basically a modern, rigorous way of restating Euler's results.
Speaking of Euler, at one point he anticipated certain aspects of the modern attitude toward divergent series. He didn't anticipate that there would be several inequivalent methods, but he did address whether or not his divergent "sums" should be called "sums", given that they sometimes looked absurd. He concluded that it didn't matter, as long as they were useful! See De seriebus divergentibus, especially sections 6 through 10. He mentions 1 + 2 + 3 + 4 + ⋯, but only in passing, as a series related to 1 + 2 + 4 + 8 + ⋯. Although he doesn't focus on 1 + 2 + 3 + 4 + ⋯, it might be possible to say something about his argument in this article. I'm not sure how to fit it in, but anyone is welcome to try! Melchoir (talk) 05:10, 2 February 2014 (UTC)

Summability > Summary part: Incorrect statement[edit]

"[...] Those methods [Cesàro Summation & Abel Summation] work only on oscillating series; they cannot produce a finite answer for a series that diverges to +∞. [...]"


This sentence makes 2 statements for 2 methods, 4 statements in total:

  • (A) Cesàro Summation works only on oscillating series.
    • (B) Thus, Cesàro Summation cannot produce a finite answer for a series that diverges to +∞.
  • (C) Abel Summation works only on oscillating series.
    • (D) Thus, Abel Summation cannot produce a finite answer for a series that diverges to +∞.

Follow-ups are all right, if A is correct, B also is, since a series that diverges to +∞ may not be oscillating, same for C and D.

However, I don't know about the C, but the statement A itself is not correct. Cesàro Summation does work for series that do not oscillate, as long as the series do not diverge. For example, it works for the series ∑(1/2)n, sequence of which starting with a1:



\begin{align}
\\S_n &=  \textstyle \sum_{k=1}^n 1/2^n
\\S_n &= 1 - 1/2^n
\\
\\CesSum &= \lim_{b \to \infty} \frac{\textstyle \sum_{n=1}^b S_n}{b}
\\CesSum &= \lim_{b \to \infty} \frac{\textstyle \sum_{n=1}^b (1-1/2^n)}{b}
\\CesSum &= \lim_{b \to \infty} \frac{b - \textstyle \sum_{n=1}^b (1/2^n)}{b}
\\CesSum &= 1 - \lim_{b \to \infty} \frac{\textstyle \sum_{n=1}^b (1/2^n)}{b}
\\CesSum &= 1 - \lim_{b \to \infty} \frac{S_b = CesSum = \mbox{with the assumption that it is bounded}}{b}
\\CesSum &= 1 - 0
\\CesSum &= 1
\end{align}


My suggestion: Adding "Excluding/aside from the convergent series," right at the beginning of that sentence, assuming that Abel thing works out for the convergent non-alternating series as well. Else, some other formulation... — manual signature: comment added by ThoAppelsin (talkcontribs) 14:39, 2 February 2014 (UTC)


Good point. Does this work? [4] Melchoir (talk) 19:20, 2 February 2014 (UTC)


It does work for me, but the critical side of me wants me to say that this may get misunderstood by people who think like a computer. "convergent series and oscillating series" does seem to imply convergent series ∨ oscillating series in logic or convergent_series | oscillating_series in C (bitwise), thanks to repeated mentioning of the word "series". Chance of misunderstanding would even be higher if it was worded as "Those methods work only on convergent and oscillating series; [...]", but I don't know, would it look bad if we just had an "or" there, or would then there be a risk of getting misunderstood as exclusive or, as in "either... or" (⊕ in logic, or ^ in C, bitwise)? Whatever, here are my thoughts as an observer; you decide, maybe and/or, to make sure that it gets understood as inclusive or...
Hmm. The thing about "or" is that some readers might misinterpret it as "or, in other words". I'm not sure which is the greater risk. Melchoir (talk) 21:27, 2 February 2014 (UTC)

argumentative[edit]

This word has been used for the reasoning of the latest edit: https://en.wikipedia.org/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&diff=593759534&oldid=593726973

In which sense "argumentative", could I have an elaboration for the reason?

Absolutely, and thanks for asking! (I meant to elaborate on the talk page, but I was distracted...)
I've seen elsewhere on the Internet that there are plenty of debates over heuristic derivations. The debates tend to center around whether or not the derivations are meaningful, and whether it is wise to let impressionable young mathematicians see them. These are subjective value judgements. I have my own opinion, as does everyone else. What we cannot do on Wikipedia is enforce an opinion on others. The change I reverted used the phrases "one has to keep in mind ... cannot be used as a definition ... has to be seen ..." (emphasis in the original). Such language is inappropriate for an encyclopedia.
Generally, we should follow the principles of the policy Wikipedia:Neutral point of view. This is an easy requirement to follow in mathematics, since there are usually precise formulations for any point of view that one might wish to represent. The more precise the language, the less we risk pushing a POV. For example:
  • Bad: This derivation is acceptable. Good: The derivation is given in Ramanujan's notebook.
  • Bad: This derivation is unacceptable. Good: The derivation is the less rigorous of his two arguments.
  • Bad: Step 1 is not justified. Good: Step 1 belongs to a family of transformations that are not always valid, and it is not justified by inclusion in this family alone.
In each example, the Bad and the Good sentence express the same sentiment. The Bad version is a bare value judgement, which is debatable. The Good version is a more objective version that should be uncontroversial.
Incidentally, I also object to the merits of the edit that I reverted. Its mathematical content was the statement that the Dirichlet series of the zeta function does not converge for the argument -1. Yes, that should be pointed out. The statement belongs in the following section, after the Dirichlet series of the zeta is introduced. And in fact, it's already there. Melchoir (talk) 19:10, 3 February 2014 (UTC)

Secondary Summary > "number of mathematically interesting results"[edit]

The second paragraph of the whole article starts with this following sentence:

"Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis, quantum field theory and string theory."

I want to draw the attention over the part between the first and second comma. Unless I am missing something, there is only one, not a number of, mathematically interesting result; which is the assertion, I must say, that 1 + 2 + 3 ... equals -1/12. That aside, I have the feeling that the adjective interesting is not entirely neutral, encyclopedic, and is redundant, similar to how despised would be.

Suggestion: "[...] it can be manipulated to yield a [[Real number|real]] result, which has [...]" — Preceding unsigned comment added by ThoAppelsin (talkcontribs) 20:34, 3 February 2014 (UTC)

Unfortunately, the author of that passage doesn't seem to be active anymore: 80.168.225.18. So they probably won't be commenting here...
I think by "result" they meant "a mathematical statement", not "a number which is the output of an algorithm". So, the fact that zeta function regularization gives you −1/12 is one result. The fact that Ramanujan summation gives you −1/12 is a second result. These two results are related, but they're not the same.
That said, the choice of words isn't perfect, especially "interesting". We already have citations supporting "famous" and "remarkable", so yes, it's redundant. The sentence is pretty vague in general. And does complex analysis really belong in the category of "other fields" when we're talking about divergent series? Melchoir (talk) 00:51, 4 February 2014 (UTC)
I don't think evaluation of the divergent series ∑n has anything to do with complex analysis, although I possibly may not be knowing what complex analysis really is. It rather is just an overcommitment to desperately assign a value to a divergent series, and just a value, which has been achieved through the exclusion of many other manipulations ending up with many other results that would be possible unless a function with terms that have variable powers, eventually to be set to 1, but disallowing that replacement until the manipulation is done, was introduced, so that truly only a single manipulation could be made. Still an invalid manipulation, but an invalid manipulation undercover, and by being the only single manipulation, despite being invalid, no contradictory result is to be found, thanks to the isolated status of it, thanks to the abettor regularized ζ.
So, yeah, nothing imaginary there, no square root of -1, just a function which accepts complex numbers, but never does in this case. It could very well be a function that is only defined for real numbers. And if this -1/12 does have an appliance in complex analysis, then I would say that the complex analysis fits well in that other category. — manual signature: comment added by ThoAppelsin (talkcontribs) 02:54, 4 February 2014 (UTC)

Linearity and stability[edit]

As far as I can tell, there are two methods that may be called zeta function regularization. If we replace

\sum_{n=1}^{\infty}a_n \to \sum_{n=1}^{\infty}a_n^{-s}

and analytically continue to s = −1, then we get a method that is stable but not linear. If we replace

\sum_{n=1}^{\infty}a_n \to \sum_{n=1}^{\infty}a_n\cdot n^{-s}

and analytically continue to s = 0, then we get a method that is linear but not stable. It would probably be better to call the first method "zeta regularization" and the second method "Dirichlet regularization", but that's just my opinion.

For the purpose of regularizing 1 + 2 + 3 + 4 + ⋯, it doesn't matter which method we choose. We wind up with ζ(−1) either way, so we can afford to be vague. But in general, it can matter, and the difference is hinted at in the Infinite arithmetic series article.

So for the new section "Failure of stable linear summation methods", is it better to say that zeta function is not stable, or that it is not linear? Or should we avoid committing to either statement? It may be easier to focus on cutoff regularization / Ramanujan summation, which is definitely linear but not stable.

...and in the time it took me to write this comment, R.e.b. has already fixed the article. :) Melchoir (talk) 01:46, 7 February 2014 (UTC)

Multiline Equations giving errors[edit]

I don't know what the problem is but instead of displaying any of the multiline equations on this page, all I'm seeing is the error "Failed to parse(unknown function '\begin'):" followed by the Latex code in red. Site bug? — Preceding unsigned comment added by 152.106.99.20 (talk) 13:44, 7 February 2014 (UTC)

Yes seem to be a site wide problem see Wikipedia talk:WikiProject Mathematics#Problem with multiline equations and Wikipedia:Village pump (technical)/Archive 123#Math aligned environments failing to parse. If you were logged in you could switch to MathJax rendering which works fine.--Salix alba (talk): 14:49, 7 February 2014 (UTC)

the "smoothed curve"[edit]

Not sure what is meant by smoothed curve. I can see that the minimum of the quadratic that fits the points (1,1), (2,3), (3,6), (4,10) etc is -1/12, but that its not the quadratic that is graphed there. I propose that you clarify the meaning of "smoothed curve" in this context. -papasandy

Thanks for the feedback! I'm sure the prose could be improved, but I'm not sure exactly which part you're referring to. There are three diagrams that show a curve, and there is some accompanying text in the "Smoothed asymptotics" section, but the phrase "smoothed curve" isn't used.
By the way, the quadratic that fits the points (1,1), (2,3), (3,6), (4,10), ... directly is x(x+1)/2, which has a y-intercept of 0 and a minimum of −1/8. The catch is that when we promote that sequence to a function of a real variable, the resulting stair-step function is jagged and does not approach a quadratic. The original quadratic x(x+1)/2 only fits the top of the steps; shifting it 1 unit to the right gets us x(x−1)/2, which fits the bottom of the steps. At each extreme, the y-intercept is 0. If we compromise and shift half a unit to the right, we get a y-intercept of −1/8, coinciding with the minimum of the quadratic. If we average over all possible shifts, we get an average y-intercept somewhere between 0 and −1/8, which is, lo and behold, \int_0^1\frac{x(x-1)}{2}=-\frac{1}{12}. Anyway, I digress, because that isn't the general definition of the smoothed sum. There might be a connection to the general theory, but I haven't attempted to prove or disprove such a connection, and in any case, I wouldn't add such material to the article without a reference.
Returning to cutoff regularization as described by Tao: After applying a cutoff to the partial sums, we do get a smooth function with a quadratic asymptote, which is −1/12 + CN2. The article explains what it means to apply a cutoff. Melchoir (talk) 00:53, 12 February 2014 (UTC)

Interpolation Image is wrong[edit]

The Intersecting values of the interpolated curve follow the function y=(x^2)/2, so how can x=0,y=-1/12 ? Unless I am mistaken, can anyone tell me why this is? Or is it part of another function? Because all the summed values follow the function y=(x^2)/2Shroobtimetraveller (talk) 07:22, 12 March 2014 (UTC)

If you're talking about Image:Sum1234Summary.svg, the green curve is y=(x^2)/2 - 1/12.
Meanwhile, the nth partial sum of the original series is (n^2 + n)/2. Melchoir (talk) 08:24, 12 March 2014 (UTC)

"Failure of stable linear summation methods" section inadequate[edit]

This section includes the following:

   0 +1 + 2 + ... = 0 + x = x by stability

Subtracting gives

   1 + 1 + 1 + ... = x – x = 0 by linearity 


This is wrong. What has been done here is that the two sequences have been subtracted term by term. This assumes some sort of term alignment, which is not possible with infinite series (see the explanation of 1+2+3+... = -1/12 further up the page).

The only acceptable description of 0+1+2+... -x is (0+1+2+...)-(1+2+3+...) or (-1/12)-(-1/12) or 0

If we start aligning terms arbitrarily, we need to remove the explanation of x=-1/12 — Preceding unsigned comment added by 81.147.10.55 (talk) 15:51, 21 April 2014 (UTC)

Minus one twelfth[edit]

NOTE: Copy of relevant information from a different page.

Hi Drbogdan,

You seemed very anxious to know whether it is "true" that \sum_{n=1}^\infty n=-\frac{1}{12}. The talk pages of those articles is not really the appropriate venue for discussing that, but it seems reasonable to drop you a note here.

The problem is that you're trying to ask whether it's "true" before it has been agreed what, if anything, it actually means. In the most usual approach to infinite series, the equation is certainly not true. One would say that \sum_{n=1}^\infty n diverges (see divergent series). Or, in the context of measure theory, one could say \sum_{n=1}^\infty n=\infty.

That said, there are certain interpretations under which the equation is true. I don't agree with the commenters who call it a "fallacy" or "nonsense". But you can't just assert it as "true" out of context, because under the most usual interpretations of the notation, it is definitely not true. --Trovatore (talk) 19:11, 5 February 2014 (UTC)

@Trovatore - Thank you *very* much for your comments above (and at Talk:Infinity#Minus one twelfth) regarding my recent edit:

Copied from the Infinity lead (updated-20140206):

Interestingly, the summation of all natural numbers to infinity is "minus one-twelfth".< ref name="NYT-20140203">Overbye, Dennis (February 3, 2014). "In the End, It All Adds Up to –1/12". New York Times. Retrieved February 3, 2014. </ref>< ref>Sondow, Jonathan Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series. Proc. Amer. Math. Soc. 120 (1994), no. 2, 421–424.</ref>

\sum_{n=1}^{\infty} n = - \frac {1}{12}
your comments are *greatly* appreciated - seems the equation may be "true" in some contexts - but not in others (esp. usual interpretations) - Thanks again for your comments - and - Enjoy! :) Drbogdan (talk) 20:02, 5 February 2014 (UTC)
See the page 1 + 2 + 3 + 4 + ⋯ where the sumation is examined in detail.--Salix alba (talk): 16:24, 6 February 2014 (UTC)

@Salix alba - Thank you for your comment - and link - yes, the notion seems well described in the article - also, somewhat related, I made the following userbox (updated to your new link)

\int \!\,
This user knows that \sum_{n=1}^{\infty} n = - \frac {1}{12}
reference


thanks again for your comment - and - Enjoy! :) Drbogdan (talk) 17:55, 6 February 2014 (UTC)