How can I prove the convergence of a power-tower? In here, I saw that $$x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$$ exists as a real number (convergent) if and only if $$x\in[e^{-e}, e^\frac{1}{e}].$$ How can I prove this??
 A: 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 :)
A: 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 $x\in[0,e^{1/e}]$

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 $y=x^y$
A: As Lucian mentioned above, I have considered the function $$f(y)=y^{\frac{1}{y}}$$ $$\lim_{y\to 0^+}f(y)=0$$   $$\lim_{y\to \infty}f(y)=1$$  Also $$f'(y)=y^{\frac{1}{y}}\Big(\dfrac{1}{y^2}-\dfrac{\ln y}{y^2}\Big).$$ Using this, we have maximum of $f(y)$ occurs at $y=e.$ Hence $$f(y)\le e^{\frac{1}{e}},\,\,\,\,\,\, \forall y\ge 0.$$ But I did not have an idea about obtaining the lower bound $e^{-e}.$
