Discussion of TPID 6

[JJacquelin]

Conjecture

\( \lim_{n\to\infty} f(n) = e^{1/e} \) where \( f(n) = x \) such that \( {}^{n}x = n \)

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 equations

• \( x = 1 \)
  
• \( x^x = 2 \)
  
• \( x^{x^x} = 3 \)
  
• \( x^{x^{x^x}} = 4 \)
  

and so on... is the sequence under discussion. The conjecture is that the limit of this sequence is \( e^{1/e} \), also known as eta (\( \eta \)). Numerical evidence indicates that this is true, as the solution for x in \( {}^{1000}x = 1000 \) 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 \( {}^{1000}x = 1000 \) is approximately 1.44467831224667 which is higher than e^(1/e)
The solution for x in \( {}^{10000}x = 10000 \) 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 \( {}^{n}x \) is e , for n tending to infinity. So, the limit isn't = n , as expected.