07/25/2010, 11:51 PM
(10/07/2009, 12:03 AM)andydude Wrote: Conjecture
where
such that
Discussion
To evaluate f at real numbers, an extension of tetration is required, but to evaluate f at positive integers, only real-valued exponentiation is needed. Thus the sequence given by the solutions of the equationsand so on... is the sequence under discussion. The conjecture is that the limit of this sequence is
, also known as eta (
). Numerical evidence indicates that this is true, as the solution for x in
is approximately 1.44.
lim n-> oo x^^n = n conj : any real x = eta
since (eta+q) ^^ n grows faster than n for any positive q , we can use the squeeze theorem
lim q -> 0 eta =< x <= eta + q
hence x = eta
see also http://en.wikipedia.org/wiki/Squeeze_theorem
QED
regards
tommy1729