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MathGroup Archive 1999

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Re: a tricky limit

  • To: mathgroup at smc.vnet.net
  • Subject: [mg15715] Re: [mg15327] a tricky limit
  • From: Jurgen Tischer <jtischer at col2.telecom.com.co>
  • Date: Fri, 5 Feb 1999 03:42:14 -0500 (EST)
  • Organization: Universidad del Valle
  • References: <199901080915.EAA03988@smc.vnet.net.>
  • Sender: owner-wri-mathgroup at wolfram.com

Arnold,
I think this is the last time I comment about your tricky limit. This
nice David Lichtblau read all my code and comments and pointed out to
me some errors I made and showed me that my upper limit in fact should
be much better as I thought. In the end there was little I had to
correct, essentially it amounts to that my corrected lower bound is
much better so here is the result:

upper bound: 2.292173695248657289096168980403 (as before)

lower bound: 2.292173695221057149738189560492 (much better)

this makes it ten digits and an error less than 3 10^-11. I wonder what
you need that thing for, but it was lots of fun.

Jurgen

If someone wants the notebook, please email me.


Arnold Knopfmacher wrote:
> 
> I wish to obtain a numerical estimate (say 8 decimal digits) of the
> limit as x  tends to 1 from below of  the function
> h[x]=(Product[(1-fm[x]/(m+1)),{m,2,Infinity}])/(1-x) where
> fm[x]=x^(m-m/d) and d is the smallest divisor of m that is greater than
> 1. The problem is that when I replace Infinity by say 1000 as the upper
> limit  of the product, the function blows up near 1. Visual inspection
> of the graph of h[x] for 0<x<0.9 say, suggests that the limit should
> have a value around 2.1. Can anyone help?
> 
> Thanks
> Arnold Knopfmacher
> Dept of Computational and Applied Math Witwatersrand University
> South Africa


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