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