On Dec 28, 1:54 pm, achrzesz <[email protected]> wrote: > On Dec 28, 6:26 am, shreevatsa <[email protected]> wrote: > > > > > Hi, > > > I'm trying to use Sage to find the asymptotics of binomial > > coefficients. Specifically, I wanted to find out the rate at which > > binomial(n, n/2)/2^n goes down to 0 as n goes to infinity. > > > See Wolfram > > Alpha:http://www.wolframalpha.com/input/?i=%28n+choose+n%2F2%29+%2F+2%5En > > which gives a "series expansion at n=∞" from which (with some manual > > work) we can find out that it is > > sqrt(2/pi) * 1/n^(1/2) - 1/(2*sqrt(2*pi)) * 1/n^(3/2) + O(1/n^(5/2)). > > (and presumably Mathematica does what Wolfram Alpha does too). > > > How to do the same thing in Sage? > > > I tried this: > > > sage: var('n') > > n > > sage: f = binomial(n, n/2) / 2^n > > sage: f(n = 4) > > 3/8 > > sage: taylor(f, n, infinity, 2) > > > --------------------------------------------------------------------------- > > TypeError Traceback (most recent > > call last) > > ... > > [snip] > > ... > > TypeError: ECL says: Error executing code in Maxima: taylor: > > encountered an unfamiliar singularity in: > > binomial(n,n/2) > > > Next, trying the trick > > athttp://doxdrum.wordpress.com/2011/02/19/sage-tip-series-expansion/ > > I tried changing n to 1/n: > > > sage: g = binomial(1/n, 1/(2*n)) / 2^(1/n) > > sage: g(n = 1/4) > > 3/8 > > sage: taylor(g, n, 0, 2) > > > --------------------------------------------------------------------------- > > TypeError Traceback (most recent > > call last) > > ... > > [snip] > > ... > > TypeError: ECL says: Error executing code in Maxima: taylor: > > encountered an unfamiliar singularity in: > > binomial(1/n,1/(2*n)) > > > Same results. Incidentally, "taylor(f, n, 0, 2)" works, but "taylor(g, > > n, infinity, 2)" doesn't. I've also tried the same with binomial(2*n, > > n) and binomial(2/n, 1/n), and even with binomial(2*n*n, n*n) and > > binomial(2/(n*n), 1/(n*n)) (for which Wolfram Alpha sort of gives a > > power series in n instead of sqrt(n)), but the same results. > > > Is there a way of getting the asymptotics of this function in Sage? > > > Thanks, > > Shreevatsa > > #NO GUARANTEE OF VALIDITY# > sage: from sympy import * > sage: n=symbols('n',integer=True) > sage: f=binomial(n,n/2)/2**n > sage: g=f.series(n,oo,3);g > pi**2*n**2*exp(-n*log(2))/24 + exp(-n*log(2)) + O(n**3) > #I think should be O(1/n**3) > sage: g1=g.removeO() > sage: limit(g1,n,oo) > 0
Hello Shreevatsa I'm affraid, I misinterpreted the sympy expansion The assymptotics of the central binomial coefficient is more subtle matter Andrzej Chrzeszczyk -- 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
