On Dec 28, 6:14 pm, achrzesz <[email protected]> wrote:
> 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
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
Andrzej Chrzeszczyk
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