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 is approximately 1.44467831224667 which is higher than e^(1/e)

The solution for x in is approximately 1.4446796588047 which is higher than e^(1/e)

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 is e , for n tending to infinity. So, the limit isn't = n , as expected.