about power towers and base change

[sheldonison]

This is still unresolved.
Despite my increasing understanding of sexp and slog, I do not know how to work with this.

Im confident others here could solve the problem.

regards

tommy1729

Traveling -- little time right now. This is problem is closely related to my mathstack thread; ln-n-of-tetrationx-n-is-nowhere-analytic.

My relevant function is the following equation.
\( f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n \)


One of Tommy's original question was
"is there a known series expansion for y^^(n-1) = x^^n ??"

For the limiting answer, as n goes to infinity, given base base x, we use my function above. We desire \( f(y)=e^{f(x)} \), so we calculate \( y=f^{-1}(e^{f(x)}) \). Of course, there is the small issue that f(x) is conjectured coo and nowhere analytic .... I plan to eventually post a proof.

Using the above equation, if b~=7.28550781987618684208203148323, from this mathstack thread
then f(b)=e, and as n goes to infinity, \( b\uparrow\uparrow n = e \uparrow \uparrow (n+1) \)
- Sheldon