Timeline for Convergent series, asymptotics and truncation
Current License: CC BY-SA 3.0
9 events
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| May 1, 2013 at 16:23 | comment | added | Gerhard Paseman | Here is a weak but related example that interests me, and might interest you. One of the answers to mathoverflow.net/questions/37679/… mentions work of Stevens, where he uses Bonferroni inequalties to justify truncating a sum to improve an upper bound. You might track examples like that to see if they lead you to where you want to end up. Gerhard "Ask Me About System Design" Paseman, 2013.05.01 | |
| May 1, 2013 at 16:15 | comment | added | Gerhard Paseman | I like the question. I just think that "of course" should be replaced by "I will assume", since I can come up with examples where your N does not exist. This is the kind of subject Knuth might treat in his Art of Computer Programming Series, and perhaps briefly in his Concrete Mathematics book. I fear that although such methods are practiced on a near daily basis, there is not enough of the subject to warrant a long chapter. If you ask about this within a discipline (e.g. sieve methods in analytic number theory) you might get a hit. Gerhard "And Also You Might Not" Paseman, 2013.05.01 | |
| May 1, 2013 at 11:39 | history | edited | Kevin Smith | CC BY-SA 3.0 |
Grammatical errors fixed, removed repeated statements
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| May 1, 2013 at 11:29 | comment | added | Kevin Smith | I should say, rather than behaviour on compact sets. | |
| May 1, 2013 at 11:24 | history | edited | Kevin Smith | CC BY-SA 3.0 |
deleted 43 characters in body; deleted 1 characters in body
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| May 1, 2013 at 11:21 | comment | added | Kevin Smith | Gerhard, thank you. One should note that the scenario I've given in the second paragraph may be taken as the definition of convergence for each $x$. Agreed, the assumptions implicit in my second displayed equation are more restrictive. I do not assume uniformity of convergence because the question is focused on growth rather than behaviour at individual points. | |
| Apr 30, 2013 at 15:35 | comment | added | Kevin Smith | Would the converse not imply there exist values of $x$ for which it is not convergent? Perhaps I'm missing something. It's not essential ($O(1)$ would be enough) but I think it is neater this way. | |
| Apr 30, 2013 at 15:13 | comment | added | Gerhard Paseman | I think your scenario following "of course" is not implied by pointwise convergence: think of f_n(x) as g(x-n), where g is a smooth approximation to the characteristic function of the unit interval. To get the tail small enough, you will need some uniformity in the shrinking of the f_n (or a lot of cancellation). Gerhard "Ask Me About System Design" Paseman, 2013.04.30 | |
| Apr 30, 2013 at 14:09 | history | asked | Kevin Smith | CC BY-SA 3.0 |