---
tag: RDC61V812J
kind: example
name: Iterated exponentiation
msc: [40A05, 26A18]
instance_of: [B88PFYXWBK]
uses: [0RMVS7RXZX, 0FM6N4R0VB, 0KN22AAW5R]
sources:
  - url: https://en.wikipedia.org/wiki/Tetration
---

## 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|B88PFYXWBK]] 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|0FM6N4R0VB]] 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.

## Explanations

### L1 human

A power tower stacks the same number in the exponent again and again.
For $\sqrt2$ the values creep up towards $2$ and never pass it, because $\sqrt2$ raised to the power $2$ is again $2$.
For larger bases the tower explodes.