Re: Another limit problem
- To: mathgroup at smc.vnet.net
- Subject: [mg67766] Re: [mg67711] Another limit problem
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Thu, 6 Jul 2006 06:54:52 -0400 (EDT)
- References: <200607050818.EAA26450@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 5 Jul 2006, at 17:18, Virgil Stokes wrote:
> I am trying to evaluate the limit of the following expression as s
> goes
> to infinity,
>
> \!\(\((1 + â??\+\(k = 0\)\%\(s - 1\)\((\((s\ Ï?)\)\^k\/\(k!\))\)/\((\
> ((s\ \
> Ï?)\)\^s\/\(\(s!\) \((1 - Ï?)\)\))\))\)\^\(-1\)\)
>
> where, Ï? (Real) < 1, s (Integer) > 0. I am quite sure that the
> limit is
> 0; but, I am unable to get this result using Mathematica 5.2.
> Any suggestions would be appreciated.
>
> --V. Stokes
>
I don't think Mathematica alone can prove this but with the help of
Mathematica I seem to be able to prove it for any numerical rho, and
probably could do so in general if I could devote a little more time
to it. But as I can't I have decided to post my incomplete argument
below, hoping that someone will either to complete the proof or find
my mistake (or perhaps a bug in Mathematica ;-)).
All we need to show that the reciprocal of your expression goes to
Infinity as s->Infinity. Mathematica gives us the following
expression in terms of incomplete Gamma:
FunctionExpand[FullSimplify[
Sum[(s*r)^k/((k!*(s*r)^s)/
(s!*(1 - r))),
{k, 0, s - 1}]]]
((-E^(r*s))*(r - 1)*s*
Gamma[s, r*s])/(r*s)^s
where I used r instead of your rho (and I ignored +1 as it does not
affect the answer). Now let's us re-write it in the form:
s!*((E^(r*s)*(1 - r))/
(r*s)^s)*(Gamma[s, r*s]/
(s - 1)!)
From the formula for the value of Gamma[n,x] where n is an integer
given here:
http://mathworld.wolfram.com/IncompleteGammaFunction.html
we see that Gamma[s, r*s]/(s - 1)!) -> 1 as s->Infinity. Therefore
our limit is the same as the limit
Limit[s!*(E^(r*s)*(1 - r) /
(r*s)^s),s->Infinity,Assumptions ->{0<r<1}]
Unfortunately Mathematica 5.1 is still unable to resolve this, but if
we substitute a numerical value for r it seems to be able to do so,
and the answers it gives seem to be as follows:
0<r<1
r = 3/4; Limit[
s!*(E^(r*s)*((1 - r)/
(r*s)^s)),
s -> Infinity,
Assumptions -> {0 < r < 1}]
Infinity
1<r
r = 2; Limit[s!*(E^(r*s)*
((1 - r)/(r*s)^s)),
s -> Infinity]
-Infinity
It suggests that your statement is actually valid for any r>0. I
think this should not be too hard to prove by hand but I have
absolutely no more time to think about it as I am leaving for Europe
tomorrow morning and must start packing now! If the problem is still
open in about one week's time I will try to return to it.
Andrzej Kozlowski
Tokyo, Japan
- References:
- Another limit problem
- From: Virgil Stokes <vs@it.uu.se>
- Another limit problem