Re: Re: Problem with limiits

*To*: mathgroup at smc.vnet.net*Subject*: [mg65855] Re: [mg65845] Re: Problem with limiits*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 19 Apr 2006 04:54:03 -0400 (EDT)*References*: <200604160544.BAA07913@smc.vnet.net> <e1vdsq$967$1@smc.vnet.net> <200604181056.GAA14316@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On 18 Apr 2006, at 19:56, Roger Bagula wrote: > Andrzej Kozlowski wrote: > >> >> For any sequence of real numbers a1,a2, a3 .... converging to a real >> number L, any infinite subsequence of it will converge to the same >> limit L. This is a very basic fact and is in fact true in any >> complete metric space (a subsequence of a convergent sequence is a >> Cauchy sequence, and therefore, in a complete metric space, is itself >> convergent.) >> >> Given the above, you only need the fact that there are infinitely >> many primes (proved by Euclid and before him by Eudoxus). >> >> Andrzej Kozlowski >> > What you are saying is that if I have a composite number c[m] > them the Limit: > > Limit[(1+1/c[m])^c[m],m->Infinity]=E > > as well. > What About: > Limit[(1-1/m)^m,m->Infinity] =1/E > > Your reasoning gives: > Limit[(1-1/c[n])^c[n],n->Infinity]*Limit[(1+1/Prime[n])^Prime[n],N- > >Infinity]=1 > and > Limit[(1-1/m)^m,m->Infinity]*Limit[(1+1/m)^m,m->Infinity]=1 > This does give: > Limit[(1 - 1/m^2)^m, m -> Infinity]=1 > > Thanks for your help. > No, I am saying that you have to have a *sub-sequence*. (1+1/Prime[n])^Prime[n] for n from 1 to Infinity is a sub-sequence of (1 + 1/m)^m from 1 to Infinity. If you can't see this general fact just look at Table[(1 + 1/Prime[n])^Prime[n], {n, 1, 3}] {9/4, 64/27, 7776/3125} and Table[(1 + 1/m)^m, {m, 1, 5}] {2, 9/4, 64/27, 625/256, 7776/3125} and it should become clear. On the other hand: (1-1/m)^m for m from 1 to Infinity is not a sub-sequence of (1 + 1/n) ^n from 1 to Infinity. Just look at the first few terms: Table[(1 - 1/m)^m, {m, 1, 5}] {0, 1/4, 8/27, 81/256, 1024/3125} Do I need to say more? Andrzej Kozlowski

**References**:**Problem with limiits***From:*Roger Bagula <rlbagulatftn@yahoo.com>

**Re: Problem with limiits***From:*Roger Bagula <rlbagulatftn@yahoo.com>