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Re: Problem with limiits
*To*: mathgroup at smc.vnet.net
*Subject*: [mg65761] Re: [mg65695] Problem with limiits
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Mon, 17 Apr 2006 02:27:59 -0400 (EDT)
*References*: <200604160544.BAA07913@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On 16 Apr 2006, at 14:44, Roger Bagula wrote:
> A well known limit is:
> Limit[(1 + 1/n)^n, n -> Infinity]=E
> I tried it and it works... solution seems built in.
>
> I tried:
> Limit[(1 + 1/Prime[n])^Prime[n], n -> Infinity]
>
> Again I tried:
> Limit[(1 + 1/Prime[n])^Prime[n], n -> 2000]
>
> Here's how I got an estimate:
> Table[(1 + 1/Prime[n])^Prime[n], {n, 1, 400}];
> ListPlot[%]
>
> It appears to be approaching E as well.
> N[(1 + 1/Prime[2000])^Prime[2000], 100] - E
> -0.0000781568388416035074355625109358426411345791124581922819707122937
> 62387821356624136556497567576200
>
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
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