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.

**Follow-Ups**:**Re: Re: Problem with limiits***From:*Daniel Lichtblau <danl@wolfram.com>