On Dec 28, 2011 9:05 PM, "shreevatsa" <[email protected]> wrote: > > On Dec 29, 3:29 am, John Cremona <[email protected]> wrote: > > >> The assymptotics of the central binomial coefficient is more subtle > > >> matter > > >> Andrzej Chrzeszczyk > > > > > The limit: > > > sage: maxima('limit(binomial(2*x,x)/2^(2*x)*sqrt(%pi*x),x,+inf)') > > > 1 > > > > > shows that > > > > > binomial(2*x,x)/2^(2*x) ~ 1/sqrt(pi*x) as x->+oo > > > > This follows very easily from Stirling's Formula. But I suppose the > > original question was not so much "what is the answer" as "can I get > > the answer from system [xyz] without having to think"! > > > Yes! > > After all, anything Sage can do a human can probably do with enough > patience and time. The point of the software is to make it easy.
(Not meant as flaimbait) I can think of a lot of counterexamples to this, at least in the context of number theory problems that involve counting, which would be impossible to do in a lifetime because humans are just way too slow... > > In this case, I would like to be able to input an expression like > binomial(n, n/2) or binomial(n, 17*n/32), and get a power series in > sqrt(n) or whatever. For instance, the Wolfram Alpha link above, after > clicking on "more terms", gives > 2^(-n) (sqrt(2/pi) sqrt(1/n)-(1/n)^(3/2)/(2 sqrt(2 pi))+(1/n)^(5/2)/ > (16 sqrt(2 pi))+(5 (1/n)^(7/2))/(64 sqrt(2 pi))-(21 (1/n)^(9/2))/(1024 > sqrt(2 pi))-(399 (1/n)^(11/2))/(4096 sqrt(2 pi))+(869 (1/n)^(13/2))/ > (32768 sqrt(2 pi))+(39325 (1/n)^(15/2))/(131072 sqrt(2 pi))-(334477 (1/ > n)^(17/2))/(4194304 sqrt(2 pi))-(28717403 (1/n)^(19/2))/(16777216 > sqrt(2 pi))+(59697183 (1/n)^(21/2))/(134217728 sqrt(2 pi))+O((1/ > n)^(23/2))) exp(log(2) n+O((1/n)^12)) > > This is beyond my patience or ability to do it without mistakes. > Exactly what computers are for! > > My belief is that if Wolfram Alpha can do it, it must be possible for > Sage to do it as well. It seems that something has not been > implemented here, but I'm not sure what. (The asymptotics for the > gamma function?) > > Actually, Wolfram Alpha doesn't realize that exp(log(2)n) and 2^-n can > be cancelled, and the expression simplified. Further, exp(O((1/n)^7)) > can be expanded as a power series. So even WA's output still requires > some human work, and there is scope for Sage to improve on it. > > -- > To post to this group, send email to [email protected] > To unsubscribe from this group, send email to [email protected] > For more options, visit this group at http://groups.google.com/group/sage-support > URL: http://www.sagemath.org -- To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
