TITLE: Values for which tetration to infinite heights (i.e., $x^{x^{x^{x^{.^{.^{.}}}}}}$) converges
QUESTION [5 upvotes]: I was reading Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$). For what values of $x$ does tetration to infinite heights  (i.e., $x^{x^{x^{x^{.^{.^{.}}}}}}$) converge?

REPLY [8 votes]: Look up "exponential tower" or "tetration".
Calculating the expression
involves Lambert's famous $W$ function
with a value of
$\frac{W(-\ln x)}{-\ln x}$.
Wikipedia has a nice discussion at
http://en.wikipedia.org/wiki/Tetration.
As shown there,
your expression converges for
$e^{-e} \le x \le e^{1/e}$.
As with at least 50% of the
questions proposed here,
this was shown by Euler.