Timeline for Lacunar series with an interesting (in-formula) symmetry.
Current License: CC BY-SA 4.0
9 events
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| Dec 30, 2018 at 18:16 | history | edited | Glorfindel | CC BY-SA 4.0 |
image fixed (inlining HTTP images doesn't work anymore); for more info, see https://gist.github.com/Glorfindel83/9d954d34385d2ac2597bbe864466259f
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| Apr 4, 2010 at 23:04 | comment | added | Charlie Frohman | Get rid of the first term of the series, and I think that mess sums absolutely in the interior of the unit disk. :) The limit as $q\rightarrow 0$ is uninterestingly $0$. In fact then the whole story is the alternating sum of $\pm q$. Depending on your method, that could add up to a lot of things. One of my favorite chapters of any book is the first chapter of Hardy's "Divergent Series". At the end of it, you are pretty sure that Hardy had lost his marbles. | |
| Apr 4, 2010 at 22:10 | comment | added | Richard Stanley | A somewhat related question appears in Enumerative Combinatorics, vol. 2, Exercise 6.63(c). | |
| Apr 4, 2010 at 22:06 | comment | added | Yemon Choi | Jonas, Reid: yes, of course, I was being dim (and failing to read the relevant expression properly). Thanks for clarifying. | |
| Apr 4, 2010 at 21:46 | comment | added | Reid Barton |
Right, it certainly might be the case that you get an interesting function by regularizing the sum $q-q+q-q+\cdots$ appropriately, but without knowing what nsum is doing, question 1 is hard to answer.
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| Apr 4, 2010 at 21:34 | comment | added | Jonas Meyer | @Yemon: It converges only when q=0; consider the terms when m=0. Abel or Cesaro means will converge, and I'm guessing that that is where the 1/2 comes from, but Reid's assessment seems fair. | |
| Apr 4, 2010 at 21:22 | comment | added | Yemon Choi | Reid: when you say it is not convergent, are you claiming it fails to converge for each q in the unit disc? or merely that it fails to converge for some q in the (closed) unit disc? | |
| Apr 4, 2010 at 20:52 | comment | added | Reid Barton |
The sum defining X(q) is not convergent. I see the documentation of nsum mentions that it will happily sum certain divergent series, but if you don't know exactly what it is doing, this means it is silently producing nonsense.
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| Apr 4, 2010 at 19:33 | history | asked | graveolensa | CC BY-SA 2.5 |