Wikipedia:Featured article candidates/1 − 2 + 3 − 4 + · · ·


 * The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article review. No further edits should be made to this page.

The article was promoted 23:04, 16 March 2007.

1 − 2 + 3 − 4 + · · ·
Self-nom. This two-week-old article has had an appearance on DYK and a brief peer review, which I'm cutting short so that I can get the ball rolling here. Other editors have made helpful suggestions, and I think the article is looking pretty good.

The article isn't very long, but I think it's comprehensive. The topic isn't very important, but I think it's interesting. The subject area isn't very accessible, but… it lends itself to pretty pictures? Melchoir 10:36, 11 March 2007 (UTC)


 * Comment ooooh, pretty pictures. --Ideogram 10:54, 11 March 2007 (UTC)


 * Comment WP:MOS says to avoid special characters in headings, specifically the slash. Not sure what to do about φ-summation&mdash;is there a way you can rename that section?  Sandy Georgia  (Talk) 14:10, 11 March 2007 (UTC)
 * Is there a missing word here? Ref 16:  ^ Knopp p.491; there appears to be an at this point in Hardy p.3   Sandy Georgia  (Talk) 14:17, 11 March 2007 (UTC)
 * Done and done. Melchoir 19:48, 11 March 2007 (UTC)


 * Comment the second paragraph of the introduction needs some work. In particular, "(1/4) is the arithmetic mean between 0 and 1⁄2, where 1⁄2 is itself the mean of 0 and 1".  This doesn't seem to relate to either the series in question, or the rest of the article. Bluap 15:47, 11 March 2007 (UTC)
 * Well, that part was meant to refer mostly to "Stability and linearity" and "Cesàro and Hölder". But if it doesn't work, that's what the delete key is for… Melchoir 19:51, 11 March 2007 (UTC)
 * …There we go. Melchoir 19:54, 11 March 2007 (UTC)
 * Comment I find it hard to muster much enthusiasm for what I regard as a rather trivial subject, but that's not related to the critieria, is it? On the other hand, I really like the inventive use of images. I do have a couple of specific comments though:
 * Section Divergence, "This sequence is notable for ...": That's a cute remark, but it does seem rather unrelated to the subject of the article.
 * It's a little unrelated, yes. But it's only one degree away; you can't have a comprehensive article on the series without including the sequence of partial sums, and if one were going to write an article on that particular sequence, you'd have to include the remark. It wouldn't make much sense to have separate articles, so we can pretend that we merged them. Melchoir 18:26, 12 March 2007 (UTC)
 * Section Stability and linearity, "Since ... does not have a sum in the usual sense, the equation ... must be given some other meaning.": Is that logically sound? Might the equation not be meaningless?
 * I actually considered saying something to the effect of "the equation is either meaningless or it must have/be given some other meaning", which is safer but sounded silly. I'll just rewrite that bit to be less active. Melchoir 18:29, 12 March 2007 (UTC)
 * Same section, "... Cesaro summation and Abel summation are just two examples": Given that the reader has not yet been introduced to these summation methods, this statement (specifically, the word just) may a bit puzzling.
 * Well, they're briefly mentioned in the lead, but I guess it's better not to depend on that. I'll fix it. Melchoir 18:34, 12 March 2007 (UTC)
 * Same section, "A different approach, also applicable to a range of method": I had to read this twice, and once more after I read the whole article, before I understood its meaning. Reformulate?
 * I had a lot of trouble with that transition when I wrote it; it's good that I'm forced to fix it now. This should do it. Melchoir 19:01, 12 March 2007 (UTC)
 * Section Euler and Borel, "So the Euler transform of ... is ...": This sentence uses a without defining it. Furthermore, the structure is not clear. The Euler transform of an arbitrary series is 1/2 + 1/4 (&Delta;a)_0 + ... and for "our" series, this evaluates to 1/2 - 1/4 = 1/4.
 * It should be better now. Melchoir 19:55, 12 March 2007 (UTC)
 * Section Separation of scales: I don't know &phi;-summation, but I find it odd that a mathematical theorem is established by using two physical principles. What's going on here?
 * It's more that the definition of the method, and the constraints one assumes of phi, are motivated by those physical principles. There's a mathematical proof of the precise statement in the book, which broadly follows the summary in the article. After the authors are done with the proof, the moral of the story is "Thus the principle of infinitesimal relaxation combined with the separation of scales condition permitted us to arrive at a striking result/ 1 − 2 + 3 − 4 + … = 1/4." I think I can make it a little clearer… Melchoir 19:08, 12 March 2007 (UTC)
 * Section Generalizations: What's the original reference containing Abel's quote? I ask because I have a source (Boyd, Acta Appl. Math. 56:1-98) which places it in 1828.
 * Grattan-Guinness writes that it is a letter "to Holmboe in January 1826", and his footnote cites "Abel Letters, p. 16; Correspondence, pp. 16-17. Mostly also in Abel Works1, vol. 2, pp. 266-267; Works2, vol. 2, pp. 256-257." Markushevich writes "Cauchy … Algebraic Analysis … And after another five years Abel wrote in a letter to his friend Holamb:". Google tells me that Cauchy's work was published in 1821, which again places Abel's letter in 1826. Melchoir 19:21, 12 March 2007 (UTC)
 * Oh, and Grattan-Guinness does have bibliographic details in pp.154-155 of the bibliography, but I'd rather not copy them all out here! Melchoir 19:24, 12 March 2007 (UTC)
 * Notes: What are the citations of "Euler et al." (notes 10, 11 and 19)? The only reference with Euler has no coauthors. On checking that reference now, it seems that some accents have gone missing.
 * The coauthors are the translators, whose notes at the end of the paper are the source for the modern interpretations in this article. I'll expand the reference. Melchoir 19:26, 12 March 2007 (UTC)
 * Hmm, looks like it was a template error to begin with. Melchoir 19:27, 12 March 2007 (UTC)
 * Oh, yes, the accents aren't in the English translation, but I see them in the original document, so I restored them.
 * Pictures: Many do not work on my browser (Debian Iceweasel 2.0.0.1); I assume that's a technical problem which should not influence this discussion. As I said, the pictures are a real addition to the article. However, they are not so easy to understand. The picture in the section Stability and linearity works perhaps better if you place the four copies of the series on top of each other. I don't understand the Euler summation picture. I assume that it's not possible to explain the pictures completely in the captions, but perhaps a more extensive explanation can be added to the picture pages. -- Jitse Niesen (talk) 11:39, 12 March 2007 (UTC)
 * That's really strange… the three SVGs on 0.999... didn't cause any problems, or at least I assume someone would have spoken up if they had. I don't think I've changed the way I generate them. If necessary, they could be made into large PNGs until the underlying technical issue is resolved.
 * The current layout in Stability and linearity had the advantage that the participants in the "staple" cancellations are right next to each other. If all four copies were stacked vertically, that part would be a lot harder to communicate.
 * Euler is a little tricky; I'll see what I can do with the Image: pages. Melchoir 19:38, 12 March 2007 (UTC)
 * I've tried jimmying a few things; if you clear your cache and reload the article, does it work now? Melchoir 20:11, 12 March 2007 (UTC)
 * I've added lengthy descriptions to the image pages, including what is probably a contender for Longest Description Ever for the Euler sum. I think I've now hit all your points. Melchoir 21:31, 12 March 2007 (UTC)
 * Thanks, that's a lot! I'll interleave comments above to make the page easier to read. Melchoir 18:20, 12 March 2007 (UTC)
 * Basically support I like the article, great pictures (although some don't some of the thumbs don't appear in my browser so this has to be fixed), nice depth, well written and concise. But if and when this becomes a featured article, please remember to avoid making it the FA of the day! The title seems pretty innocuous but the article is bound to be beyond confusing for anyone who has forgotten (or never learned) calculus. Even in the lead, a sentence like "Euler treats these two as special cases of 1 − 2^n + 3^n − 4^n + · · · for arbitrary n" will probably lose a few readers, not to mention the rather counterintuitive idea that one might assign a sum to a sum that doesn't sum... In that line of thought, I'm not sure that it's worth wikilinking "Solving the equations": if anyone has gotten that far, they know what solving an equation means. Also I think the lead needs to expand a bit on the significance of this series (historically and as an example illustrating the various summation methods) which is what the rest of the article is really outlining. Pascal.Tesson 07:17, 13 March 2007 (UTC)
 * Thanks for the comments! I'll… start with the short ones.
 * Stupid thumbnails… I really thought I'd fixed them this time. I'll have to seek help on the Village Pump…
 * I don't think the third paragraph of the lead is so bad.
 * If anything, I was worried that the lead made the series sound too important! Doesn't it touch on the main themes already?
 * Equation solving might not be that useful, but it's conceivable that someone will want to click on it. Its presence there is also kind of a signal that that section should be accessible in a limited sense: even if the readers don't understand the deeper concepts, at least they can go through the motions with the article as their guide.
 * Actually, that philosophy works throughout the article. Even if you (the generic you) don't really get what's going on, someone who's taken a high-school course in calculus will see a lot of familiar concepts: a Taylor series here, a limit there… look, with a little effort I can even do that integral! An enterprising high-school teacher might even use the article to construct a final test: see if you can compute all of the following quantities. Is there a pattern? Of course, an introductory college course would be even more helpful, and there are plenty of people who have taken one of those: engineers, bio majors, and so forth.
 * Finally, I'd like to say that I think it's important for an article like this to get to the Main Page. Granted, it's probably more demanding than any article that has been Today's Featured Article thus far. Maybe not by much, given Enzyme inhibitor and Photon, but still. Its precedent would encourage other editors to showcase technical topics, which is good for the encyclopedia as well as its public image. There's a certain attitude out there that Wikipedia is for pop culture, and I think some of the recent TFAs should already be dispelling that notion; this one would really help. Internally, we could always use more activity in the sciences, and that's hardly just mathematics; Featured article candidates/Irish phonology needs love too! We may see a time when there are two FAs side-by-side every day, and it could even become a tradition that one should be general-interest and the other more specialist. But I'd rather not wait for that day; I'd rather be actively encouraging it! Melchoir 16:10, 13 March 2007 (UTC)
 * I'm giving you bonus points for sheer faith! Thumbnails are now fine and slight changes in the lead have been quite good so my main reservations have been taken care of. Good work Melchoir. Pascal.Tesson 14:03, 15 March 2007 (UTC)
 * Thanks! I've been reading Going Postal, which emphasizes how one can gain favor by offering hope to others. (Well… it's a lot more shady than that, but we'll leave it there.) Melchoir 21:42, 15 March 2007 (UTC)


 * Support, but please consider the use of non-breaking hard spaces in the equation, which often breaks up on my screen. Although not required by MOS, using nbsp will keep the equation (1 − 2 + 3 − 4 + · · ·) intact on one line. Sandy Georgia  (Talk) 14:56, 13 March 2007 (UTC)
 * I used to use non-breaking spaces, but I now use nowrap, which is reportedly even more effective and has the added bonus of keeping the markup easy to read. I just added 11 more that were missing, so there shouldn't be any more problems. The only places where I haven't put in nowraps are footnotes, captions, and the beginnings of paragraphs. Melchoir 15:18, 13 March 2007 (UTC)


 * Comment Just so everyone knows, I've switched all of the SVGs to PNG versions. They're a little harder to read, but they're reliable. Melchoir 17:45, 13 March 2007 (UTC)


 * Support Nice article. Comment on the lede: the second paragraph seems to jump into talking about summations right away - it's not entirely clear why this is so relevant that it's almost the first thing talked about. Maybe the third paragraph should come before the second, because the end of the third paragraph gives us some idea of why the series is relevant (any more information to this end would be a welcome addition as well). MarkBuckles (talk) 08:11, 14 March 2007 (UTC)
 * Maybe a transition should be made into the second paragraph, but I'm not sure that the series' relevance to research is the way to do it. For an article on Eta constants, that would be appropriate, but this article shouldn't claim more territory than it's entitled to. The series 1 − 2 + 3 − 4 + · · · all by itself is relevant to the study of divergent series because it is such a nice example. It is probably the second-simplest of all divergent series, 1 − 1 + 1 − 1 + · · · being the first. The problem is that I wouldn't want to write the preceding sentence in the article, because it's kind of POV. If you care more about form than ease of summation, you might say that 1 + 2 + 3 + 4 + · · · is the second-simplest divergent series after 1 + 1 + 1 + 1 + · · ·. And Euler's own classification of divergent series into four categories doesn't reflect either of these opinions. Anyway, despite my rambling, I'll try to improve the transition. Melchoir 19:20, 14 March 2007 (UTC)
 * Is this better? Melchoir 19:25, 14 March 2007 (UTC)
 * It flows better but the rhetorical questioning "How could this be?" seems out of place for an encyclopedia (even though that's probably what everyone's thinking). MarkBuckles (talk) 20:13, 14 March 2007 (UTC)
 * True… how about this? Melchoir 20:24, 14 March 2007 (UTC)
 * Tried my hand too . Also I'm confused about the syntax in this sentence: "For many summability methods, 1 − 2 + 3 − 4 + · · · is an easy example, and they assign it a "sum" of 1⁄4 after all." What's the predicate? Is that and supposed to be there? MarkBuckles (talk) 23:47, 14 March 2007 (UTC)
 * I rearranged that sentence, and I filled in a dangling "this" right after the equation. Melchoir 00:03, 15 March 2007 (UTC)
 * By the way, given Euler's words as quoted later on, including "no more doubtful" and "incontestably", I'm pretty sure that Euler did feel justified in writing the equation. I agree that we don't need it, though. Melchoir 00:06, 15 March 2007 (UTC)
 * Do not support on the basis of the article's title. Please see my comments on the article's talk page. Robert K S 09:47, 14 March 2007 (UTC)
 * Support - The images are help understanding a lot. We need more mathematics FA's. Leon math 21:07, 15 March 2007 (UTC)
 * Comment - How come this article appears in the Featured Articles list (under the mathematics section), but doesn't have a star on it? (The FA list seems to be the only source that says this article is an FA) Leon math 23:25, 16 March 2007 (UTC)
 * I think you caught Raul in the middle of an update. Melchoir 23:28, 16 March 2007 (UTC)
 * Or, more precisely, GimmeBot hasn't closed the FAC yet. I guess the only thing it doesn't do is add the star, so I'll do that now. Melchoir 23:31, 16 March 2007 (UTC)
 * The above discussion is preserved as an archive. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article review. No further edits should be made to this page.