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 at http://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 -- 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
