Hi
I played a bit around with the behave of approximation when the tetrate progresses to its fixpoint and the stepwith for iteration is increasing, say exponentially, for instance at step k=3 the iteration-height for x_3 is j=2^3=8 and at step k=4 the iteration-height for x_4 is 2^4=16 and so on. Then how do the x_(k+1)/x_k -ratios behave?
Clearly this seems to approximate 1; but I modified the criterion a bit; I used
\( \hspace{24} \log(\frac{ x_{\small k+1}}{x_{\small k}}) ^{\frac1{\small 2^k}} \)
and find, that I get the log of the fixpoint with this, at least for some tested bases.
Formally:
with \( b=t^{\frac1t} \), t in the range 1<t<exp(1)
it seems that
\( \hspace{24} \lim_{k->\infty, \ j=2^k} \hspace{24} \log( \frac{b\^\^ ^{2j}}{b\^\^ ^j })^{\frac1j} -> \log(t) \)
Perhaps there is an "obvious" reason, which I overlooked...
Gottfried
I played a bit around with the behave of approximation when the tetrate progresses to its fixpoint and the stepwith for iteration is increasing, say exponentially, for instance at step k=3 the iteration-height for x_3 is j=2^3=8 and at step k=4 the iteration-height for x_4 is 2^4=16 and so on. Then how do the x_(k+1)/x_k -ratios behave?
Clearly this seems to approximate 1; but I modified the criterion a bit; I used
\( \hspace{24} \log(\frac{ x_{\small k+1}}{x_{\small k}}) ^{\frac1{\small 2^k}} \)
and find, that I get the log of the fixpoint with this, at least for some tested bases.
Formally:
with \( b=t^{\frac1t} \), t in the range 1<t<exp(1)
it seems that
\( \hspace{24} \lim_{k->\infty, \ j=2^k} \hspace{24} \log( \frac{b\^\^ ^{2j}}{b\^\^ ^j })^{\frac1j} -> \log(t) \)
Perhaps there is an "obvious" reason, which I overlooked...
Gottfried
Gottfried Helms, Kassel