For $x>0$, let $\tau_0 = 1$ and $\tau_{n+1} = x^{\tau_n}$. The infinite power tower of $x$ is then $\tau = \tau(x) = \lim_{n\to \infty} \tau_n$. It is well known that $\tau$ exists and is finite for $x \in \left[\frac{1}{e^e}, e^{\frac{1}{e}}\right]$. What I am wondering is:
- How can one show $\lim_n \tau_n = \infty$ for $x > e^{\frac{1}{e}}$? It seems true from numerical experiments but should it be possible to prove this using elementary estimates? It is trivial by induction if $x \ge 2$ but I'm not sure how to push the proof up to $e^{\frac{1}{e}}$.
- Intuitively, it is clear that if $x$ is really small, since $x^x \to 1$ as $x\to 0$, then $\tau_n$ should oscillate between $x$ and $1$. Is it true that if $x < \frac{1}{e^e}$, then there are two distinct positive real numbers $a=a(x), b=b(x)$ such that if $(\tau_{n_k})$ is a convergent sub-sequence of $(\tau_n)$, then $\lim_k \tau_{n_k} \in \left\{a, b\right\}$? Again, numerical experiments suggest so but I don't know how to proceed for the general proof.
Any hint is appreciated!