Convergence of $\ln\frac{x}{\ln\frac{x}{\ln x...}}$

Question

I don't really know how to work with continued fractions, so I'm not too sure where to start. But this is my question: I've got a function:

$$y=\ln\frac{x}{\ln\frac{x}{\ln x...}}\tag{a}$$

I try rewriting a closed form of (a) as:

$$y=\ln{\frac{x}{y}}\tag{b}$$

When I graph this though, it seems that (a) doesn't converge to all values of (b). Specifically, when graphed, it seems that (a) diverges at around x=e. I'm not sure how to show that however, would someone be able to explain?

Answer 1

Let $f(t) = \ln(x/t)$. For $x > 0$, the fixed point of the function $f$ is $W(x)$, where $W$ is the Lambert W function. Now $$f'(W(x)) = -\frac{1}{W(x)}$$ This is always negative for $x>0$, and has absolute value $< 1$, implying the fixed point is stable, if and only if $x > e$. Thus for $x > e$ it is possible that the limit is in fact $W(x)$, while for $0 < x < e$ it is not possible.