tiny limit-curiosity [ from ratio b^^(2j) / b^^j ]

[tommy1729]

Gottfried wrote :

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 think you can reduce to :

\( \hspace{24} \lim_{k->\infty, \ j=2^k} \hspace{24} \log( \frac{h(b)}{b\^\^ ^j })^{\frac1j} -> \log(t) \)


it seems to converge faster and to the same value.

for instance dont you get the same result for the ratio b^^3j / b^^ j as the ratio b^^2j / b^^ j ?

it seems so at first sight , without using many iterations though.

im not sure im right here , but you have the power to check it.

btw i mainly tried base sqrt(2) for convenience.

high regards

tommy1729