Talk:1 + 2 + 4 + 8 + ⋯

message: 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)
thread_title: To do

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  s = 1 + 2s, s = infinity 

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

message: 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)
replies: text: 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)
replies: text:  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)

text: 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)

text: Even if the encyclopedia is a work in progress? Melchoir 09:50, 13 May 2007 (UTC)
replies: text: 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)
thread_title:  merge suggestion 

message: 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)
replies: text: 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)
thread_title:  Confusing title 

message: 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)
thread_title:  Function value 

message: 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!
* http://www.math.dartmouth.edu/~euler/pages/E247.html}}
** In History of Grandi's series on 2011-03-29 14:29:08, 404 Not Found
** In 1 + 2 + 4 + 8 + … on 2011-05-25 06:48:48, 404 Not Found
** In 1 + 2 + 4 + 8 + … on 2011-06-09 12:23:10, 404 Not Found
--JeffGBot (talk) 12:23, 9 June 2011 (UTC)
thread_title:  Dead link 

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