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