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 tetration limit ?? tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 05/29/2011, 07:28 PM (This post was last modified: 05/29/2011, 07:45 PM by tommy1729.) (04/02/2009, 09:56 PM)bo198214 Wrote: Perhaps then you should start with the simpler case of the double iterate. And look what a suitable function f you would find that: $\lim_{n\to\infty} (1+f(n))^{(1+f(n))^{n}} = Q$ I dont see what useful function f that could be. we reduce to (1+1/f(n))^((1+1/f(n))^n) = Q in essense we only need to understand the relation between n and f(n). further switch f(n) and n to get (1+1/n)^((1+1/n)^f(n)) = Q ln(1+1/n) * (1+1/n)^f(n) = ln(Q) replace ln(Q) by Q f(n) = ln(Q/ln(1+1/n)) / ln(1+1/n) done. but lim n-> oo sexp_(1+f(n))[slog_(1+f(n))[n] + 1/2] = n + C 0 < C seems harder and not so related at first. worse , it might have problems stating it like above ... because our n needs to be after the second fixpoint and our superfuntions need to be defined at their second fixpoint ... which " evaporates " at oo as n goes to oo. and hence our superfunctions become valid and defined > q_n where lim q_n = oo !! if f(n) does not grow to fast this might be ok , but on the other hand to arrive at C at our RHS f(n) seems to need some fast growing rate. so f(n) is strongly restricted and C must be unique and existance is just assumed. i do not know anything efficient to compute f(n) apart from numerical *curve-fitting* upper and lower bounds as described above. or another example , actually the original OP rewritten : lim n-> oo sexp_(1+f(n))[slog_(1+f(n))[n] + 1/2] = C 0 < C now we must take the first fixpoint approaching 1 .. or the second ?? it seems easiest to take the first fixpoint , if we take the second we have the same problem of the " evaporating ' fixpoint as above. on the other hand , we dont know the radiuses for bases 1+f(n) expanded at their first or second fixpoint. again , its hard to find f(n) and C despite they are probably strongly restriced - even unique -. another idea that might make sense is that there exists a function g(n) such that but lim n-> oo sexp_(1+f(n))[slog_(1+f(n))[n] + 1/2] = g(n) 0 < g(n) < n and that g(n) gets closer and closer towards the end of the radius of one of the fixpoint expansions as n grows. and that might be inconsistant with the other equations/ideas above. so many questions. regards tommy1729 « Next Oldest | Next Newest »

 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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