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 tetration limit ?? gent999 Junior Fellow Posts: 5 Threads: 1 Joined: Apr 2009 04/15/2009, 04:41 PM (This post was last modified: 04/15/2009, 06:25 PM by gent999.) bo198214 Wrote:I just wanted to know whether these are wellknown facts (and only i didnt hear of them yet) or whether it are observations on the research front. Because I think only these two facts are already noticable alone. And it would desirable to have proper proofs for them. A more thorough investigation of these cases, $\alpha>1$ $\lim_{n\to\infty} (e^{1/e}+n^{-\alpha})[4]n$ could perhaps also provide ideas for your other questions. Did you experiment with them? Can you plot a graph in dependence of $1<\alpha<2$? Or is the limit $e$ for all $\alpha>1$? And what about other fixed points, e.g. $d=\sqrt{2}$? These calculations were made several years ago but currently my notes in another country. I can try to restore them, if it is interestring. Thereat I considered function: c(n,x) = d + x ^-(1+s), s>0 and wrote Taylor expansion for n = 1, 2, 3. Comparing expansions I wrote recurrent expressions for first coefficients and using induction proved them for all n. Thus, constant part was d[4]n, linear part was reduced (due to selection of d) and to investigate the question, quadratic part (quite bulky) and order of residue was sufficient. The conclusion was that for any s>0 the sequence converges to e. Numeric experiment was fulfilled in Mathematica, for s=1 convergence confirmed. It is interesting, that the numeric process has two stages: first, when resulting value grows (the more, the less is s), and second, where it is converges. I’ve tried several 0

 Messages In This Thread tetration limit ?? - by tommy1729 - 04/01/2009, 05:49 PM RE: tetration limit ?? - by nuninho1980 - 04/01/2009, 08:15 PM RE: tetration limit ?? - by bo198214 - 04/02/2009, 09:58 PM RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 12:53 AM RE: tetration limit ?? - by bo198214 - 04/03/2009, 12:49 PM RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 05:54 PM RE: tetration limit ?? - by bo198214 - 04/02/2009, 02:50 PM RE: tetration limit ?? - by tommy1729 - 04/02/2009, 09:24 PM RE: tetration limit ?? - by bo198214 - 04/02/2009, 09:56 PM RE: tetration limit ?? - by tommy1729 - 04/02/2009, 10:39 PM RE: tetration limit ?? - by tommy1729 - 05/29/2011, 07:28 PM RE: tetration limit ?? - by bo198214 - 05/31/2011, 10:34 AM RE: tetration limit ?? - by nuninho1980 - 04/03/2009, 06:12 PM RE: tetration limit ?? - by bo198214 - 04/06/2009, 10:49 PM RE: tetration limit ?? - by nuninho1980 - 04/07/2009, 01:35 AM Updated results for tetration limit - by sheldonison - 10/31/2010, 03:32 PM RE: tetration limit ?? - by nuninho1980 - 04/04/2009, 02:21 PM RE: tetration limit ?? - by gent999 - 04/14/2009, 10:12 PM RE: tetration limit ?? - by bo198214 - 04/14/2009, 10:31 PM RE: tetration limit ?? - by gent999 - 04/15/2009, 12:18 AM RE: tetration limit ?? - by bo198214 - 04/15/2009, 01:35 PM RE: tetration limit ?? - by tommy1729 - 04/15/2009, 03:05 PM RE: tetration limit ?? - by gent999 - 04/15/2009, 04:41 PM RE: tetration limit ?? - by tommy1729 - 04/29/2009, 01:08 PM RE: tetration limit ?? - by BenStandeven - 04/30/2009, 11:29 PM RE: tetration limit ?? - by tommy1729 - 04/30/2009, 11:38 PM RE: tetration limit ?? - by BenStandeven - 05/01/2009, 01:35 AM RE: tetration limit ?? - by BenStandeven - 05/01/2009, 01:00 AM RE: tetration limit ?? - by JmsNxn - 04/14/2011, 08:17 PM RE: tetration limit ?? - by tommy1729 - 05/28/2011, 12:28 PM RE: tetration limit ?? - by nuninho1980 - 10/31/2010, 10:31 PM RE: tetration limit ?? - by JmsNxn - 05/29/2011, 02:06 AM RE: tetration limit ?? - by tommy1729 - 05/14/2015, 08:29 PM RE: tetration limit ?? - by tommy1729 - 05/14/2015, 08:33 PM RE: tetration limit ?? - by tommy1729 - 05/28/2015, 11:32 PM RE: tetration limit ?? - by sheldonison - 06/11/2015, 10:27 AM RE: tetration limit ?? - by sheldonison - 06/15/2015, 01:00 AM RE: tetration limit ?? - by tommy1729 - 06/01/2015, 02:04 AM RE: tetration limit ?? - by tommy1729 - 06/11/2015, 08:25 AM

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