Iterated exponentiation

Example

For $x > 0$, the infinite power tower $x^{x^{x^{\cdots}}}$ is defined as the limit of $t_1 = x$, $t_{n+1} = x^{t_n}$. It is an infinite composition of the map $t \mapsto x^t$. Take $x = \sqrt2$; the tower converges to $2$.

Check: $t_1 = \sqrt2 < 2$, and if $t_n < 2$ then $t_{n+1} = \sqrt2^{\,t_n} < \sqrt2^{\,2} = 2$. Since $t \mapsto \sqrt2^{\,t}$ is increasing and $t_2 = \sqrt2^{\sqrt2} > \sqrt2 = t_1$, the sequence is increasing. By the monotone convergence theorem it has a limit $L \le 2$ with $\sqrt2^{\,L} = L$. The equation $\sqrt2^{\,L} = L$ has the two solutions $L = 2$ and $L = 4$, and $L \le 2$ gives $L = 2$.

A classical result, going back to Euler, is that the tower converges exactly for $e^{-e} \le x \le e^{1/e}$. For example, $x = 2 > e^{1/e} \approx 1.444$ gives $2, 4, 16, 65536, \dots$, which tends to infinity.