Re: Another limit problem
- To: mathgroup at smc.vnet.net
- Subject: [mg67767] Re: [mg67711] Another limit problem
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Thu, 6 Jul 2006 06:54:53 -0400 (EDT)
- References: <200607050818.EAA26450@smc.vnet.net> <443A4967-EC98-468B-9015-86ACE77D7987@mimuw.edu.pl>
- Sender: owner-wri-mathgroup at wolfram.com
Immediately after posting this I realized I had done something silly
(which however does not affect the main point). There is completely
no need to use Mathematica's FullSimplify and FunctionExpand and the
reference to the MathWorld site to get to the expression
Limit[s!*(E^(r*s)*(1 - r) /
(r*s)^s),s->Infinity,Assumptions ->{0<r<1}]
In fact this can be done by hand just by looking at the original
problem as posted by Virgil. The rest of what I wrote, I think, is O.K.
Andrzej Kozlowski
On 6 Jul 2006, at 10:51, Andrzej Kozlowski wrote:
>
> 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