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MathGroup Archive 2006

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Re: Problem with limiits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65819] Re: Problem with limiits
  • From: Roger Bagula <rlbagulatftn at yahoo.com>
  • Date: Mon, 17 Apr 2006 02:29:15 -0400 (EDT)
  • References: <e1snmb$860$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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
> -0.000078156838841603507435562510935842641134579112458192281970712293762387821356624136556497567576200
> 
It appears that another Limit exists that behaves in the same  way:
http://mathworld.wolfram.com/Primorial.html
> The primorial satisfies the unexpected limit
> lim_(n->infty)(p_n#)^(1/p_n)==e	(3)
> 
> (Ruiz 1997; Pruitt), where e is the usual base of the natural logarithm. 


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