Talk:1 + 2 + 4 + 8 + ⋯

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 Field: Analysis

New title?[edit]

The title seems pretty bad...it probably doesn't confirm to Wikipedia Conventions. Maybe we should find a new one? User:RideABicycle/Signature 03:34, 17 February 2007 (UTC)

I know what you mean. When I wrote the article on 1 − 1 + 1 − 1 + · · ·, I titled it Grandi's series (with redirects), even though this name is not usually used outside of historical contexts, and even there it's not standard, having such competitors as "Leibniz's series". So on neutrality, simplicity, and symmetry principles, I would have liked to call it "1 − 1 + 1 − 1 + · · ·"; it's just that that title would have been inconvenient and unconventional.
But… in this case, the series doesn't seem to have any proper name. The (only two) sources I've consulted so far just call it 1 + 2 + 4 + 8 + · · ·. I guess we could call it something like "Sum of the powers of two", but this phrase has the disadvantage that no one uses it, and in some sense it neglects the very important ordering of the series… Melchoir 04:49, 17 February 2007 (UTC)
What could happen is somebody can change the name to "Sum of the powers of two" and add a redirect link using the current title. I know it isn't ideal, but it is better than the current title.

Ojay123 (TalkE-MailContribsSandbox) 22:04, 20 August 2009 (UTC)

Another plausible title would be binary geometric series. —David Eppstein (talk) 22:20, 20 August 2009 (UTC)
I think it's a great title! So is 1 − 1 + 1 − 1 + · · ·. Crasshopper (talk) 02:26, 15 May 2015 (UTC)

To do[edit]

I'm going to nominate this article at T:TDYK now, and I'm not going to be around much for a few days. So I should acknowledge what the article currently lacks, in case it draws criticism.

  • A quick application of Aitken's delta-squared process would provide a second method producing negative one, taking some of the pressure off Euler.
  • Diluting the series with zeros between the terms: what happens? For at least one case, no original research is needed. What about other variations?
  • Stuff about the 2-adic numbers. I've found it frustrating to locate a high-quality source in this area, but it's fairly straightforward.
  • History! It's what makes this series special among all the other divergent geometric series of positive terms, so it justifies focusing on just the one. I know there's material in Hardy, and I'm guessing that I've seen more material in the same references as those at History of Grandi's series.

Melchoir 05:56, 17 February 2007 (UTC)

Comment.[edit]

When I saw this on the DYK proposals page, I initially assumed that this sequence = -1 thanks to interpreting a finite-length binary representation in two's complement notation. Would this factoid be appropriate for the article, or is that just too coincidental? SnowFire 19:42, 17 February 2007 (UTC)

Oh, it's not a coincidence at all; it's a result of the fact that a 2-adic number is completely described by the behavior of its finite-length truncations. So since 1=-1 mod 2, 11=-1 mod 2^2, 111=-1 mod 2^3, and so on, roughly speaking …111=-1 mod 2^infinity. This observation kind of mirrors the observation that you can get the p-adics either by asking for the convergence of certain sequences, or by taking a certain algebraic inverse limit.
So I think this would be a worthy addition to what David Eppstein already wrote about the convergence interpretation. I just don't know of a source to which I could attribute the material, so that the reader knows it's kosher and not black magic. Melchoir 21:17, 19 February 2007 (UTC)
One reference could be HAKMEM, which refers to interpreting this sequence in two's complement notation as equal to -1. But this is already discussed in Two's complement#Two.27s complement and universal algebra. Should we discuss it again in this article, or merely link to that section of that article? --68.0.124.33 (talk) 17:56, 30 June 2010 (UTC)

s = 1 + 2s, s = -1[edit]

Are there some better sources for this, the step from: s = 1 + 2 + 4 + 8 + .... to: s = 1 + 2 ( 1 + 2 + 4 + 8 + ... ), is mathematically incorrect. It is pre-assuming that 1+2+4+8 must go to infinity, as this is the only way 1+2+4+8 could equal 1 + 2 (1+2+4+8...), thus arriving at the only other root of s = -1, instead of properly applying the conditions to prove the correct root. I am sure others have gone over this since Hardy in 1949! --155.144.251.120 23:59, 20 February 2007 (UTC)

Not sure I understand what you mean. Expanding the expression s = 1 + 2 ( 1 + 2 + 4 + 8 + ... ) gives s = 1 + 2 + 4 + 8 + ..., by multiplying out the rest of the sequence by two. There is nothing "mathematically incorrect" about an algebraic expansion to show equality, unless there is something very basic here that I am missing. 128.211.194.107 01:20, 21 February 2007 (UTC)
There is. An equation such as s = 1 + 2 + ... is only mathematically valid if the expression on the right side of the equals sign represents an actual number. Since 1 + 2 + 4 + ... is not a convergent series, it is not valid to say, in essence, "but if it were a number, we can call it s and show that it must be -1". Frankly, I'd recommend this section (or perhaps the entire article) be moved to the power series article as an example of how you cannot trust the summation of a power series outside its radius of convergence even if the series is defined at points outside that radius.MatthewDaly 01:40, 21 February 2007 (UTC)
There are lots of mathematically valid ways of assigning actual numbers to divergent series. The only trouble is that these are not taught in school, since they draw on a firm understanding of the simpler case of convergent series, and definitions of generalized summation methods tend to be phrased in terms of the convergent sum. The upshot of which is that most people are only aware of the usual sum, and they are prone to interpreting statements about divergent series as if they were implying convergence. I think this problem is already adequately described in the paragraph beginning "The above manipulation might be called on". Melchoir 01:15, 23 February 2007 (UTC)
As a reader who felt the need to comment on this point, I would have to reassert that for me it was not adequately described. Frankly, I still don't know if this is intended as some semi-amusing "1 = -1"-esque proof or if there is some obscure topological group structure extending the real numbers in which it there is a useful application to concluding that the sum of a strictly isotone positive sequence converges at all. Perhaps you might see fit to write (or link to) an article on (E)-summations so that those of us who haven't studied it can see what illumination it provides to an understanding of sequences. Aside from that, this conclusion comes across as the sort of esoteric "gotcha" that makes many laymen consider mathematics to be intellectually unfathomable, which would seem to be to mathematic's detriment. MatthewDaly 19:00, 23 February 2007 (UTC)
Maybe not the real numbers, but there is certainly a topological group extending the integers in which the series converges: the 2-adic integers. It isn't an ordered group, though, so 1 + 2 + 4 + 8 + · · · is no longer a series of positive terms in that context. This is probably the most important application of the series, since it is connected to the series …111 and the representation of negative one on computers.
And then, there are the senses in which the divergent series of real numbers can be assigned a sum: it's not a gotcha, because these are useful in physical applications. (But I don't know what application might involve this particular series.) (E)-summation isn't the only relevant concept, as you can see by browsing this talk page. Unfortunately, I wouldn't feel comfortable writing an article on the summation method with just Hardy as a reference, because I've found that book to assume a lot of context I don't have. For example, he describes the Dirichlet eta function without ever naming it – perhaps it wasn't named at the time – so it's a good thing I didn't try to write a new article on that without doing further research.
Your last point resonates, I think, with some of what I wrote at Grandi's series in education. I guess that stuff should be mentioned in this article, since Sierpińska's findings are directly relevant to the series. On the other hand, I am beginning to think that such material should be collected in a higher-level article, so that future individual articles on divergent series and geometric series can link to it. Melchoir 20:02, 23 February 2007 (UTC)
The point is that the left side and the right side are not just equal (as you say, that makes little sense) but identical: the same sequence of terms in the same order. Perhaps that could be expressed more clearly in the article, but it is no more mathematically incorrect than the rest of the article. —David Eppstein 02:36, 21 February 2007 (UTC)
I'm no mathematician, but it seems like the right side of the equation now contains one more term than the left side. Perhaps I don't understand series well enough? Xiner (talk, email) 02:44, 21 February 2007 (UTC)
Which term do you think is missing? That's a serious question: if each term on the left is also on the right, then they are the same. Don't let the fact that both are infinite confuse you. —David Eppstein 02:47, 21 February 2007 (UTC)
s = 1 + 2 + 4 + 8 + .... is equivalent to s = 1 + 2 ( 1 + 2 + 4 + ... ), which my non-mathematical mind can't equate to s = 1 + 2 ( 1 + 2 + 4 + 8 + ... ). For example, if they were the same, then you could repeat the parenthetication forever, to s = 1 + 2 + 4 ( 1 + 2 ... ), to s = 1 + 2 + 4 + 8 ( 1 + ... ), to s*s? Xiner (talk, email) 02:57, 21 February 2007 (UTC)
Alright, let me try again. s = 1+2s does not hold for any finite expansion of the series. In particular, if we let x^2 be the last term in s, then the difference in the two sides of the equal sign is 2 x^2. And it doesn't become smaller as the series is expanded indefinitely. x approaches infinity, and x^2, well, even more so? Xiner (talk, email) 14:09, 21 February 2007 (UTC)
It's admitted that the series is a divergent series in the lead section and again in the first paragraph of "Summation", which is currently the only body section. Therefore we can't expect things to become smaller as some index approaches infinity. The interesting question is: given that we are robbed of all the nice, comforting properties of convergent series, what can we say about 1 + 2 + 4 + 8 + · · · anyway? Melchoir 01:10, 23 February 2007 (UTC)

s = 1 + 2s, s = infinity[edit]

I'm wondering about the statement in the article that s=infinity can be considered as a solution of the equation. The first obvious thing I'd imagine would be generalizing to the hyperreal numbers, but in that system, there is no infinite solution. In what sense is there are infinite solution? I can imagine saying that both sides of the equation approach infinity as s approaches infinity, but that's completely different from suggesting that there is some number system in which there is an infinite solution.--76.81.164.27 04:45, 21 February 2007 (UTC)

Hardy almost certainly had the Riemann sphere in mind. The map (z -> 1 + 2z) is a Möbius transformation, so right away we know it has two fixed points on the Riemann sphere. One of these is infinity, and the other is negative one. This is standard stuff, so feel free to explain within the article if necessary. Melchoir 01:06, 23 February 2007 (UTC)
The problem is that infinity is a fixed point of every map z -> c + rz, for finite c and nonzero r, so this gives infinity as an alternate solution of every geometric series. It seems a little less remarkable that way. I wish someone could tell us what Hardy actually said about this. – Dan Hoey 23:35, 17 April 2007 (UTC)

Abel summation[edit]

Under Summation the article implies that 1+2+4+8+... is not Cesàro summable or Abel summable. I understand why it is not Cesàro summable. But the method of summation in the following paragraph, which turns the series into a Taylor series and then sets x to 1, looks similar to Abel summation. How does it fail to be Abel summation ? Is it because x=1 is not on the boundary of the convergence disk for this particular Taylor series ? Gandalf61 11:06, 17 April 2007 (UTC)

Yes. In more detail, Abel summation takes the limit x->1, which isn't possible here. Melchoir 18:35, 17 April 2007 (UTC)

merge suggestion[edit]

Since this series is the canonical example of divergence, why don't we merge it with that page and then make this page disambig to fixed point Riemann sphere and divergent series.--Cronholm144 22:35, 12 May 2007 (UTC)

I don't agree with the merge proposal. I am not quite clear which page this article is supposed to be merged into, but if the proposal is to merge it into the divergent geometric series page, then that page mentions this series and three other important examples of divergent series, each of which has its own page. The main page would become bloated if we merged all four example back into it. Gandalf61 08:37, 13 May 2007 (UTC)
I think at least temporarily, there is a case to be made for some reorganisation of this material. I have commented more fully on this at 1+2+3+4+ .... However, I note that Grandi's series should clearly not be merged with divergent geometric series, whereas the anecdote at 1 + 1 + 1 + 1 + · · · is more appropriately discussed in the zeta function context. Consequently, there are only really two articles to include here. I do not think they would bloat the article, which is only 2718 bytes long a present — a baby by wikipedia standards ;) Geometry guy 00:04, 14 May 2007 (UTC)

Sorry I was vague, my comment was made in haste, perhaps the creation of a subpage of divergent series like important examples of divergent series could be created. Then instead of having 4 or 5 small start class articles we would have one nice B class or above containing the aforementioned series and their uses--Cronholm144 08:42, 13 May 2007 (UTC)

The existence of this article doesn't prevent anyone from writing List of divergent series. In fact, if no one ever does create that list, I'll probably do it myself someday! Melchoir 09:28, 13 May 2007 (UTC)

my intent is not to anger you, I am not a deletionist and don't want you to take my comments personally. It is just that I view articles on divergent series with a skeptical eye. If they are well written and demonstrate that they are important I am fine, but if they are almost stubs I just fail to see their need for inclusion as a part of an encyclopedia--Cronholm144 09:35, 13 May 2007 (UTC)

Is the work in progress? I'm not sure that these multiple branchings are appropriate yet--Cronholm144 09:56, 13 May 2007 (UTC)

Even if the encyclopedia is a work in progress? Melchoir 09:50, 13 May 2007 (UTC)
As well as being work in progress, it is also work in use. I have commented more fully at 1+2+3+4+ .... Geometry guy 00:04, 14 May 2007 (UTC)

Partial sum?[edit]

In the article, I read "The partial sums of 1 + 2 + 4 + 8 + … are 1, 3, 7, 15, …" but it seems to me that the digit 1 as a sum is mistaken. It is the sum of 0+1, but here the first sum is 1+2, so the series of "partial sums" should begin with 3. Since I am not a mathematician, I dare not modifiying the article, but I remark here my objection. --149.132.190.175 (talk) 10:47, 6 September 2010 (UTC)

The article is correct. The partial sums sn of a sequence xm are defined as:
s_n = \sum_{m=1}^n x_m
If we take the upper limit n to be 1 then we have
s_1 = \sum_{m=1}^1 x_m = x_1
so the first partial sum is the first term of the underlying sequence. In this case the article correctly says that the sequence of partial sums starts with 1. Gandalf61 (talk) 11:56, 6 September 2010 (UTC)
OK. Thank you for the explanation. --93.32.52.147 (talk) 12:08, 7 September 2010 (UTC)

Confusing title[edit]

Probably people with a low IQ or not used to maths will find the title "1 + 2 + 4 + 8 + ..." confusing, as it doesn't clarify what are we adding next. Pair numbers? Is 10 the next or 12? What do the ellipsis mean? 1 + 2 + 4 + 8 + ... could also be followed by 16 and 31, as in the Circle division by chords progression. Really, I would rather prefer "Addition of the powers of two", which isn't ambiguous.--Ssola (talk) 16:42, 6 October 2010 (UTC)

You can always create a redirect page. Sum of the powers of two or infinite sum of the powers of two is more grammatical. Gandalf61 (talk) 12:42, 7 October 2010 (UTC)

Function value[edit]

Why isn't f(y) = 2^(x-1) in this article anywhere? It defines the series in positive values. This is a perfect example of where wikipedia doesn't give the simplest explanation for something for no apparent reason. 67.169.49.52 (talk) 23:20, 1 June 2011 (UTC)

Dead link[edit]

During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case!

--JeffGBot (talk) 12:23, 9 June 2011 (UTC)

Regular?[edit]

"...is not totally regular..." What does "regular" mean in this context? Crasshopper (talk) 02:24, 15 May 2015 (UTC)

1+1+2+5+14+42+...[edit]

The Euler heuristic for 1 + 2 + 4 + 8 + 16+... = -1 is

\begin{array}{rcl}
s & = &\displaystyle 1+2+4+8+\cdots \\[1em]
  & = &\displaystyle 1+2(1+2+4+8+\cdots) \\[1em]
  & = &\displaystyle 1+2s
\end{array}

There is an interpretation for this sum regarding binary words. A binary word on{a,b} either is null or it starts with a or it starts with b : S = 1 + S + S

The sum of Catalan numbers has a similar heuristic, following somehow operations with binary trees :

\begin{array}{rcl}
C      & = &\displaystyle 0+1+1+2+5+14+42+\cdots \\[1em]
C*C    & = &\displaystyle 0+0+1+2+5+14+42+\cdots \\[1em]
1      & = &\displaystyle 0+1+0+0+0+0+0+\cdots \\[1em]
C*C+1  & = &\displaystyle 0+1+1+2+5+14+42+\cdots = C 
\end{array}

so the sum is

C = \frac{1 \pm i\sqrt{3}}  {2}

the nice part is that after further manipulations (product, sum, left-right shifting) one can get :

 C^7 = C

I have translated in terms of divergent series the material here : http://arxiv.org/pdf/math/0212377v1.pdf and here

http://arxiv.org/pdf/math/9405205v1.pdf that is not about divergent series. Nevertheless, manipulating binary trees reflects in manipulating the divergent series above.

Here is my question : does this make some sense to you ? Here, unlike other real "sums", the complex "sum" of a series is useful to further calculus. Is this the next level of heuristics of divergent series ? thanks :)Nboykou (talk) 00:13, 8 July 2015 (UTC)