The question is about the following:
Let a function be defined such that
$$f_n(x) = x \uparrow \uparrow n$$
Where $n$ is a natural number
Now, it is reported at many places that the function
$$F_1(x) = \lim_{n \rightarrow \infty} f_n(x)$$
Is defined only for $x \in [e^{-e} ,e ^{\frac {1}{e}}]$
For bigger $x$, the limit doesn't exist because the 'power tower' is not 'convergent' , it just keeps getting bigger.
For small $x$ , however, this is not the case. I tried to approximate this function for $x = 0.05$ on a calculator and did some 20-30 iterations. And the value seemed to oscillate between $0.137$ and $0.663$.
(Which is strange because $0.05^x = x$ has a solution approximately at $0.3502$)
So, I have two questions:
$(1)$ Is it true that the 'power tower' oscillates between two distinct limiting values or is it converging to a single value very slowly?
$(2)$ If $y = F_1(x)$ then for $x \in [e^{-e} , e^{\frac {1}{e}}]$
We can write $y=x^y$
So, for $x < e^{-e}$ , what implicit/explicit relations exist for the following functions:
$$F_2(x) = \lim_{n \rightarrow \infty} f_{2n+1} (x)$$
And
$$F_3(x) = \lim_{n \rightarrow \infty} f_{2n} (x)$$
Also, it would be appreciated if anyone can provide some intuition (preferably without involving the Lambert function) about why the lower bound of $e^{-e}$ exists.
(For example, the 'intuition' for the upper bound of $e^{\frac {1}{e}}$ comes from the fact that this is (almost) $x=y^{\frac {1}{y}}$ and $y^{\frac {1}{y}}$ attains a maximum value of $e^{\frac {1}{e}}$ which is evident from equating the derivative to zero)
Best Answer
While searching online for relevant articles on infinite exponentiation I came across this one by R Arthur Knoebel
http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3087&pf=1
It seems to answer almost all my questions so I decided to put it here as an answer for anyone else to see too.
I would like to point out that I still have one unanswered question that is not addressed in this paper. We know that for $x \in [e^{-e} , e^{1/e}]$ $$F_1(x) = \frac {W (- \ln x)}{- \ln x}$$ Where $W$ is the Lambert function
Similar to this, what is the explicit expression for $F_2(x), F_3(x)$ ? The paper even provides an infinite sum for $F_1(x)$ $$F_1(x) = \sum_{n=0}^{\infty} \frac {(n+1)^n (\ln x)^n}{(n+1)!}$$ Can some infinite sum be generated for $F_2(x) , F_3(x)$ also?