Wikipedia:Peer review/1 − 2 + 3 − 4 + · · ·/archive1

1 − 2 + 3 − 4 + · · ·
I'd like comments on everything except the lead section, which I haven't really worked on yet. Assuming all goes well, I'd like to nominate for WP:FAC as soon as possible. Melchoir 01:51, 8 March 2007 (UTC)

Oh, and if anyone understands Borel summation well enough, could you suggest what a convincing illustration would look like? Melchoir 02:40, 8 March 2007 (UTC)


 * The article seems okay to me and it has some enjoyable aspects. However it pre-supposes some mathematical knowledge on the part of the reader. I wonder whether technical articles of this nature need some sort of "prerequisite" template that lists the knowledge required to understand the content? For the Borel sum illustration, perhaps a plot of the function $$e^{-2x}(1 - x)$$ would suffice (with the enclosed area shaded)? Otherwise I'm not sure. Some of the other illustrations are not showing up in my IE browser, for whatever reason. Thanks. &mdash; RJH (talk) 15:49, 8 March 2007 (UTC)


 * A prerequisite template might be a good idea, but I was hoping that its purpose could be served by careful writing and well-placed links. Is there a particular area of knowledge that you think gets assumed before it is introduced and linked to?
 * I tried playing with a plot like that for the Borel sum, but I couldn't figure out how to make the connection with the original series.
 * SVGs sometimes appear blank to me for a day or so after they're uploaded, but the current ones ought to be stable. Is there a pattern to the ones that don't show up? Melchoir 19:47, 8 March 2007 (UTC)


 * As someone with little background in math, I'll tell you some segments that were unclear to me:
 * The quotation from Euler under "Divergence" is not obviously relevant. Is it supposed to be a segue into the next section?
 * Under "Stability and linearity": "...the series 1 − 2 + 3 − 4 + · · · can be expressed as a transformation of itself". What does that mean?
 * When you introduce n and m (Under "Cesàro/Hölder") you might want to define them for clarity ("where n is some integer"?).
 * Good luck. Don't see too many math FAs. -- bcasterline • talk 21:10, 8 March 2007 (UTC)


 * Thanks! There are cultural reasons why one doesn't see too many math FAs; I'm hoping that a few examples will prime the pump for more.
 * I'll think about better ways to use Euler's quotation.
 * That sentence isn't supposed to have a mathematical meaning; it just sets up the next few equations.
 * Yeah, I'll fix that.
 * Melchoir 22:10, 8 March 2007 (UTC)
 * …There, hope that helps. Melchoir 22:24, 8 March 2007 (UTC)


 * If the expression "transformation of itself" is not familiar to people without a background in math, you could try using terms that would be more familiar to more people. For example, use "difference" rather than "transformation".  I tried to change your paragraph to make it more clear, and you can adopt as many or as few of the changes as you like.  I'm pasting my version below.  User:khollings 9 March 2007

 The series 1 - 2 + 3 - 4 +. . . does not have a convergent sum, but the following argument shows that if it had a convergent sum, the sum should be 1/4.

The series s = (1 - 2 + 3 - 4 + ...) can be expressed as the difference of two series: 1) the series h = (1 - 1 + 1 - 1 + ...), and 2) the series s = (0 + 1 - 2 + 3 - 4):


 * {|border="0" cellpadding="0" cellspacing="0"


 * s ||= 1 − 2 + 3 − 4 + · · ·
 * ||= (1 − 1 + 1 − 1 + · · · ) + (0 − 1 + 2 − 3 + · · · )
 * ||= (1 − 1 + 1 − 1 + · · · ) - (0 + 1 - 2 + 3 - · · · )
 * ||= h - s,
 * }
 * ||= (1 − 1 + 1 − 1 + · · · ) - (0 + 1 - 2 + 3 - · · · )
 * ||= h - s,
 * }
 * }

The series, h, can be written as:
 * {|border="0" cellpadding="0" cellspacing="0"


 * h ||= 1 − 1 + 1 − 1 + · · ·
 * ||= 1 − (1 − 1 + 1 − · · · )
 * ||= 1 - h.
 * }
 * ||= 1 - h.
 * }
 * }

Solving the equations $$s = h - s$$ and $$h = 1 - h$$ yields h = 1⁄2 and s = (1⁄2)h = 1⁄4.


 * Well… I know the section is titled Heuristics, but I'm not real comfortable with saying that "if it had a convergent sum, the sum should be 1/4". On the one hand, it's not mathematically meaningful, but it is the kind of sentiment with which people often speak of divergent series, and I could probably find a source for a similar statement. Would it enlighten or confuse the reader more? Anyone else?
 * Also, I've been intending s and h to stand for ordinary numbers. The notation is admittedly confusing, but there are problems with interpreting the equations among s and h as equations among series per se. Melchoir 23:03, 9 March 2007 (UTC)


 * I've tried this. How important do you think the extra step for (1 − 1 + 1 − 1 + · · · ) - (0 + 1 - 2 + 3 - · · · ) is? Melchoir 23:13, 9 March 2007 (UTC)

I'm going to de-list this Peer Review now in favor of a FAC. If anyone has more to say, especially if I asked you a question above, please drop by Featured article candidates/1 − 2 + 3 − 4 + · · · or Talk:1 − 2 + 3 − 4 + · · ·! Melchoir 10:38, 11 March 2007 (UTC)