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. 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
