Get a closed form of an expression

Question

I try to get a closed form of the following function $f(x)$.

$a_0\left(x\right)=x$

$a_{n+1}\left(x\right) = x^{a_n\left(x\right)}$

e.g. $a_{3}\left(x\right) = x^{ \left( x^{ \left( x^x \right) } \right) }$

The function is: $f\left(x\right)=\displaystyle\sum_{k=0}^{\infty}a_{k}\left(x\right)x^{k}$

I think it is defined for $e^{-e}<x<1$, because of: https://en.wikipedia.org/wiki/Tetration#Extension_to_infinite_heights https://en.wikipedia.org/wiki/Geometric_series#Formula

Thank you for any hints

bet regards

Kevin

Answer 1

Hint: You must have $y^y = y$, where $y=\lim_{n\rightarrow\infty}{a_n(x)}$.