Infinite Power Tower

Question

I've been having fun with the problem of finding the values of $n$ for which the infinite power tower $$\sqrt{2}^{\sqrt{2}^{...^{\sqrt{2}^n}}}$$ Has a finite value. My final answer was that it converged to a finite number for $n\leq4$. I reasoned that whenever $\sqrt2$ was raised to a power between $1$ and $2$, the result would also be between $1$ and $2$, so if a value occurs anywhere along the height of the tower, the whole thing would end up being between $1$ and $2$ (converging to 2). However, now that I'm trying the same problem with a power tower of $\sqrt3$, I can't determine when it converges because $\sqrt3$ raised to a power between $1$ and $2$ can be greater than $2$. I suspect that it may not ever converge, but how do I prove this? Help?

Answer 1

If your sequence converges, the limit must necessarily be a solution to $$ (\sqrt 3)^L = L $$ But this equation has no real solution -- just plot $(\sqrt 3)^x-x$ and see that it is always positive.

($B^L=L$ has a solution for $L$ if and only if $0<B\le e^{1/e}\approx 1.44467$).

Answer 2

There isn't any need to put that much effort into finding the solution. We see that since it is a self repeating process of operations, we can give the infinite tower a value of $x$. We derive,

$x = \sqrt{3}^{x}$

$x^2 = 3^x$

After this step, the value of x can be approximated.