I came across this problem that I would like to ask you about:
For which values $a>0$ does there $\exists$ a limit of the sequence $$a, a^{a},a^{a^{a}}, a^{a^{a^{a}}}...$$
Well this looks like a recursive sequence. If $a=exp(\lambda)$ then $z_{0}=0, z_{n+1}=\lambda exp(z_{n}).$
and I guess for some $\lambda$ $\exists$ limit. I'm just not sure how to show for which.
After some trying out my idea was to check for the border case $$a> e^{1/e} $$
and $$a \leq e^{1/e}$$
but I guess both diverge. Any idea or input is greatly appreciated! Thanks