tetration limit ??

[tommy1729]

I just wanted to know whether these are wellknown facts (and only i didnt hear of them yet) or whether it are observations on the research front.
Because I think only these two facts are already noticable alone.
And it would desirable to have proper proofs for them.
A more thorough investigation of these cases, \( \alpha>1 \)
\( \lim_{n\to\infty} (e^{1/e}+n^{-\alpha})[4]n \)
could perhaps also provide ideas for your other questions.
Did you experiment with them? Can you plot a graph in dependence of \( 1<\alpha<2 \)? Or is the limit \( e \) for all \( \alpha>1 \)?
And what about other fixed points, e.g. \( d=\sqrt{2} \)?

i basicly had the same idea with [4] ( i assume -> tetration is intended ) being the half - iterate of exp(n) instead.


regards

tommy1729