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tetration limit ??
#33
(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:


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
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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
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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