(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