07/24/2009, 01:19 PM

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

and find, that I get the log of the fixpoint with this, at least for some tested bases.

Formally:

with , t in the range 1<t<exp(1)

it seems that

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

and find, that I get the log of the fixpoint with this, at least for some tested bases.

Formally:

with , t in the range 1<t<exp(1)

it seems that

Perhaps there is an "obvious" reason, which I overlooked...

Gottfried