Let $(a_n)_{n\in\mathbb{N}}$ be a series in $\mathbb{C}$ or $\mathbb{R}$. Which contraints must $(a_n)$ match to make $b_n := a_1^{a_2^{...^{a_n}}}$ converge for $n\rightarrow\infty$?
For constant series with $e^{-e}
Is this the constraint that is to be applied? That there exists an $N\in\mathbb{N}$ such that $\forall n>N: a_n\in(e^{-e},e^{1/e})$? Is there any literature about this? Potence towers are pretty interesting ;)
EDIT: The question has still not been solved. However, we were able to prove that my Idea was wrong (which was pretty easy): $5^{1^{5^{1^...}}}$ converges. (For any arbitrary number, 5 is just an example.) Further more, it has to be assumed that a maximum of 1 $a_n$ is equal to $0$ (for obvious reasons aswell)
When using logarithms to unify the Potwnce tower, using $a^{b}=e^{\log{a}*b}$ to reduce $a_1^{a_2^{a_3^{...}}}$ to $e^{\log{a_1}*a_2^{a_3^{...}}}$ and so on, it is required that the remaining term (strongly) converges to $0$ in order to keep the resulting term $e^{e^{e^{...}}}$ finite.
Doesnt anyone know about this? I will badger people at Uni with this question soon ^.^