How can I prove the convergence of a power-tower?

The equation

$$x^{x^{x\cdots}}$$

is not well defined, we have to find some way to define it to find its value.

There are two sensible definitions:

The first is to say that the thing in the exponent is actually the same as the entire thing, which is written as

$$y=x^y$$

And solving for $y$ here would give the solution.

To solve this we need a special multivalued function called the Lambert $\newcommand{\W}{\operatorname{W}}\W$ function, which satisfies $z=\W(z)\cdot\exp(\W(z))$ for all complex $z$. The solution is then

$$y=-\frac{\W(-\log x)}{-\log x}$$

If we restrict attention to real-valued $\W$, the function is defined only for $x \ge −1/e$, and is double-valued on $(−1/e, 0)$.

Since the argument is $-\log x$, which is a decreasing, and $-\log(e^{1/e})=-\frac1e$, our solution is only defined when $x\le e^{1/e}$ and when the logarithm is defined. Therefore, under this definition, the function is defined (not necessarily uniquely) and real when $x\in[0,e^{1/e}]$

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There is however another definition:

Define the sequence $a_n$

$$a_0=x\qquad\qquad a_n=x^{a_{n-1}}$$

And the value were looking for is

$$\lim_{n\to\infty}a_n$$

Which can be written as $$\lim_{n\to\infty}x\uparrow\uparrow n$$

In Nicks answer, it is explained why this is the same as the definition $y=x^y$

Using Knuth's up-arrow notation,

$$\newcommand{\W}{\operatorname{W}}x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}} = \lim_{n \to \infty} x\uparrow\uparrow n$$

Let $\displaystyle y = \lim_{n \to \infty} x\uparrow\uparrow n$ and let us assume that it does converge.

Then, $\displaystyle x^y = \lim_{n \to \infty} x \uparrow\uparrow(n+1)$

As $n \to \infty$, $n + 1 \to \infty$ (Roughly speaking $\infty \pm n \to \infty$)

Hence, $\displaystyle \lim_{n \to \infty} x\uparrow\uparrow n \equiv \lim_{n \to \infty} x \uparrow\uparrow(n+1) \implies \boxed{\displaystyle y = x^y}$

A trivial solution here is $(x,y) = (1,1)$

The variables $x$ and $y$ in $y = x^y$ are not separable (we cannot isolate the any of the variables here on to any one side ) in this equation by usual algebraic manipulations.

However, we can use the Lambert $\W$ function to isolate the variables, $$y = - \frac{\W(-\ln x)}{\ln x}$$

Considering the real values of $\W$ and Using the fixed point iteration method,
it can be seen that $\displaystyle x\in\left[\frac1{e^{e}}, e^\frac{1}{e}\right]$

Good Luck :)