 1:58 PM
the infinite tetration of $x$ converges for $e^{-e} \lt x \le e^{1/e}$
will infinite pentation of $x$ converge for any range of $x$ too? 2:24 PM

https://math.stackexchange.com/questions/3362111/the-riemann-sum-of-the-definite-integral-problem-with-a-twist

3 hours later… 5:32 PM
@Mathphile Does something like $$1.5\uparrow \uparrow \uparrow 7$$ make sense at all ? 6:25 PM
@Peter Are you looking for this en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation ?
Also I know your message wasn't intended for me but I thought I could help 7:03 PM
@Vepir I noticed your last question. How far did you check $a_k(n)$ and $b_k(n)$ ? $k=0\dots,10$ and $n=2,\dots,12$ If I'm not mistaken @Peter I guess you have closed forms containing the logarithm and the floor function. Do you think we can simplify this even more ? I'm not sure
I found similar sequences on OEIS for $k=0,1$ with similar formulas listed. I tried plugging in the $(k\pm 1)$ term inside, and as a pleasant surprise, they work seem to work for first ten values of $n$ and $k$ ($100$ terms).
So far. So, you do not have an approved formula for every k and n yet ? 7:19 PM
The two formulas presented appear to work for every $n,k$.
That (the proposed identities) is what I'm looking to prove hopefully.
They do seem simple enough and to make sense. We have number bases $n,n+1$, and the formulas both just have a quotient of their logs times the $k$ parameter which is the "digit difference". OK I think the proof is rather simple and intuitive but I'm not seeing it.
I think it would be interesting to look at the same problem, but with multiple bases $n,n+1,n+2,\dots,n+r$ and try to generalize the expressions in that direction.

4 hours later… 11:02 PM
@Peter now that I think about it I guess not
@TheSimpliFire can you make sense of $1.5\uparrow \uparrow \uparrow 7$? No, because $1.5\uparrow\uparrow1.5^{1.5}$ isn't an integer.
We want that number of copies of $1.5\uparrow\uparrow\uparrow6$ or something like that
cheerio for now 11:24 PM
@TheSimpliFire cya
hmm
can we find a closed form for $\sum_{n=1}^{\infty} \frac{1}{^nx}$ or $\sum_{n=1}^{\infty} \frac{(-1)^n}{^nx}$
where $x \gt e^{1/e}$?