# Does the infinite power tower converge for all $1>x>0$

I know that if

$$y=x^{x^{x^{x^{x\dots}}}}$$

then

$$x=y^\frac1y$$

for values of $x$ where the infinite power tower converges, so when $x\le e^\frac1e$. However, when I put the power tower into Desmos, it seems to stop being accurate at around $x\approx0.1$, no matter how many $x$'s I add.

If it is truly infinite, is it accurate all the way to 0, or does it stop? How can it be proven either way?

• You need $x\ge \dfrac{1}{e^e}$ as well as the restriction you pointed out. Perhaps "$\approx 0.1$" is close enough to this limit that the software bugs out. – David Peterson Aug 22 '16 at 2:59

## 1 Answer

For $0<x<e^{-e}$, it diverges. Note that for such small values of $x$, we get $t>x^{x^t}$ for when $t>y$ and $t<x^{x^t}$ when $t<y$, where $y=x^y$. In other words, adding more powers of $x$ ends up pushing it farther and farther away from $y$ instead of closer. Particularly, it approaches $0$ and $1$, for even and odd powers of $x$.