10/22/2010, 11:27 AM
(This post was last modified: 10/22/2010, 11:32 AM by JJacquelin.)
(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.
I think that the conjecture is false.
First, the numerical computation have to be carried out with much more precision.
The solution for x in
The solution for x in
As n increases, x increasses very slowly.
But, in any case, x is higher than e^(1/e) = 1.44466786100977
Second, on a more theoretical viewpoint, if x=e^(1/e), the limit of