about power towers and base change

[tommy1729]

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)=f(x)+1, so we calculate \( y=f^{-1}(f(x)+1) \). 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)=f(e)+1, and as n goes to infinity, \( b\uparrow\uparrow n = e \uparrow \uparrow (n+1) \)
- Sheldon

I wonder where you are travelling too.
I have seen your mathstack thread. Its intresting and reminds me of base change.

In fact isnt your function f(x) just the base change ?

Could it be you made a typo ?

Your value of 7.28 does not agree with my 6.65.
I used paper and you probably a computer.
But still I wonder, are we talking about the same things and if so , why is there such a difference ?

Thanks for your intrest.

I dont have much time either.

regards

tommy1729