bo198214 Wrote: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. \( d=\sqrt{2} \)?
These calculations were made several years ago but currently my notes in another country. I can try to restore them, if it is interestring. Thereat I considered function: c(n,x) = d + x ^-(1+s), s>0 and wrote Taylor expansion for n = 1, 2, 3. Comparing expansions I wrote recurrent expressions for first coefficients and using induction proved them for all n. Thus, constant part was d[4]n, linear part was reduced (due to selection of d) and to investigate the question, quadratic part (quite bulky) and order of residue was sufficient. The conclusion was that for any s>0 the sequence converges to e. Numeric experiment was fulfilled in Mathematica, for s=1 convergence confirmed. It is interesting, that the numeric process has two stages: first, when resulting value grows (the more, the less is s), and second, where it is converges. I’ve tried several 0<s<1, but, in most cases, calculation was overflowed on first stage.
Fixed points. I do not made any accurate analytical investigation of possible sequences which are not converge to d[4]inf, 0<=d<e^(1/e). Case n^-s, 0<s leads to expected value d[n]inf.
Interesting to look also at sequence (d+1/ (i*(n-i+1)))[4]n|i=1..n.
