Talk:1/4 + 1/16 + 1/64 + 1/256 + ⋯

Deletion
It is notable because it is one of the first infinite series to be summed.

Also, other articles with similar titles (for proof of precedent):


 * 1/2 − 1/4 + 1/8 − 1/16 + · · ·
 * 1 − 2 + 4 − 8 + · · ·
 * 1 − 1 + 1 − 1 + · · · (But this is a redirect to Grandi's series. See its talk page for discussion).
 * 1 + 2 + 3 + 4 + · · ·


 * Note: There is also 1/2 + 1/4 + 1/8 + 1/16 + · · · but I have created this myself. -- Ķĩřβȳ ♥  Ťįɱé  Ø  19:11, 15 March 2007 (UTC)

Questions: Thanks. --Finngall 19:18, 15 March 2007 (UTC)
 * Why does this deserve its own entry, rather than a mention under a more general article on infinite series?
 * Who in their right mind would go searching for an article under this name?

Answers:
 * As explained in the article, this series is notable because it is one of the first infinite series to be summed.
 * I don't see how this is relevant. But this the official way to designate articles on series, when the series lacks a common name. -- Ķĩřβȳ ♥  Ťįɱé  Ø  19:24, 15 March 2007 (UTC)

It seems to me that if a series isn't significant enough to have a more common (and more easily searchable) name, it's not significant enough to merit its own entry. But that's just me. Finngall 19:30, 15 March 2007 (UTC)


 * I've responded to that point on the AfD, but I'd also like to say that plenty of important, notable mathematical concepts don't have English names. Melchoir 18:57, 16 March 2007 (UTC)

The name may be complex, but that's simply not relevant to the notability of the article. We also have


 * 0.999...
 * "All You Zombies—"

etc. -- Ķĩřβȳ ♥  Ťįɱé  Ø  19:46, 15 March 2007 (UTC)


 * We're learning this in calculus right now. If I'm not mistaken, this is a special power series (x+x^2+x^3...). In this case x=1/4. Power series of this type converge and have the sum of (1/(1-x))-1. In this case the sum would be 1/3.Bless sins 22:21, 15 March 2007 (UTC)
 * This article should exist similar to 1 + 2 + 3 + 4 + · · ·. If we have to we can merge all notable series into a super article (Famous series) about all of them.Bless sins

For the record, I agree with the proposed deletion. This article is about a straightforward convergent infinite geometric series, which does not deserve an article in its own right - at best it could be an example in geometric series. But if there is going to be a debate about this, let's take it to AfD - that's what the process is for. Gandalf61 10:09, 16 March 2007 (UTC)

To do
I'll be away for a few days, so if this hits DYK, I admit that there's more material to add to the article. For example, the partial sum argument could be explicitly expanded out, and something could be said about 0.010101… in base 2 and 0.111… in base 4. Melchoir 08:13, 23 March 2007 (UTC)

Oh, and the lead image needs to be tarted up a lot. Melchoir 08:28, 23 March 2007 (UTC)

The number of pieces
As I see it there are three pieces, one white, one black and one gray, each with the same area (1/3) of the large square. The whole point of the figure is to show that the ratio between the area of the big square and the area of the sequence of smaller squares is 1:3, hence the three equal pieces which make up the whole large square. demo 13:30, 28 March 2007 (UTC)


 * Those three regions aren't similar in the sense of geometry. It's the fact that there are four regions, all similar to the whole, that allows one to assign three of them unique colors and to repeat the process on the remainder. Melchoir 02:57, 29 March 2007 (UTC)


 * They're not similar, but each one represents the sum. &mdash; Arthur Rubin |  (talk) 03:00, 29 March 2007 (UTC)
 * After reading up on geometric similarity I see the point. But I still think that sentence should make things clearer than it does.. demo 13:17, 29 March 2007 (UTC)
 * It is divided into three parts. The first part is the black squares, second part is gray, and third part is white.-- Ķĩřβȳ ♥  ♥  ♥  Ťįɱé  Ø  03:03, 29 March 2007 (UTC)


 * All true statements; as long as no one edits the article to claim that a square can be partitioned into three squares, I'm happy. Melchoir 03:04, 29 March 2007 (UTC)


 * OOPS, Melchoir is correct. There are 3 pieces here, but squares and triangles can be divided into 4 similar pieces, which allows the construction.  &mdash; Arthur Rubin |  (talk) 03:07, 29 March 2007 (UTC)


 * Thanks! Melchoir 03:09, 29 March 2007 (UTC)

Awful proofs
Some visual demonstrations may be attractive but are pretty awfully inefficient. I have added a simple elegant proof missing in this article just as in the 1/2+1/4+1/8+1/16+... article. Cheers 8.25.32.37 (talk) 10:29, 10 May 2013 (UTC)
 * That's not particularly simple or elegant, just as it isn't in the other article. — Arthur Rubin  (talk) 20:14, 11 May 2013 (UTC)

First, about simplicity, if you have a simpler demonstration please provide it. Second, your personal taste does not matter. Third, the article lacks a stand alone proof8.25.32.37 (talk) 01:55, 13 May 2013 (UTC)
 * There's a better version of the same proof in effect at the end in the section 'The limit'. There's no need to duplicate it. The article on geometric series can do the proper proof as really any of the things here and your proof are more in the way of plausibility arguments as they don't deal with limits properly. Dmcq (talk) 16:09, 17 May 2013 (UTC)

I disagree. The proof you are providing is indirect and cumbersome. There is no need to refer to a formal limit theory to calculate the value of this series.72.37.134.11 (talk) 14:17, 19 May 2013 (UTC)
 * ->WP:CONSENSUS. I'm going to ask for this page to be protected from ip edits. See WP:DISPUTE if you want to go any further. Dmcq (talk) 14:27, 19 May 2013 (UTC)

Rating
I've used the template to rate this article. It's important to the history of maths, particularly as it documents a method of Archimedes that is an early precursor of the infinitesimal methods of Leibniz and Newton. And it also has current relevance to the creation of mipmaps, through which much of what you see displayed on screens today is processed by the hardware on your computer or phone. But I do wonder whether there's a better title than any arithmetic expression can give …. yoyo (talk) 21:51, 30 September 2018 (UTC)