02/02/2009, 06:54 PM
Hi folks -
I've just asked this question in news:sci.math; it is a tiny question and possibly answered anywhere here around ( I didn't follow the superroot-discussion intensely) so maybe we have a link already...
Ok, let's go:
Let's define the n'th iterative root ("srt") via
as one inverse of f, returning a base if a number and a iteration-count is given, such that, for instance
and consider the sequence
Then: what is x in
The sequence decreases from 3 down to e^(1/e) + eps but I think, it cannot fall below.
On the other hand, it should arrive at 3^(1/3)...
Do I actually overlook something and the sequence can indeed cross e^(1/e)?
<urrks>
Gottfried
I've just asked this question in news:sci.math; it is a tiny question and possibly answered anywhere here around ( I didn't follow the superroot-discussion intensely) so maybe we have a link already...
Ok, let's go:
Let's define the n'th iterative root ("srt") via
Code:
f(x,1) = x f(x,2) = x^x f(x,3) = x^(x^x) f(x,k) = ...
Code:
srt(y,3) = x --> f(x,3) = y
Code:
srt(3,1) , srt(3,2), srt(3,3),..., srt(3,k),... (for k=1 ... inf )
Then: what is x in
Code:
x = lim {k->inf} srt(3,k)
The sequence decreases from 3 down to e^(1/e) + eps but I think, it cannot fall below.
Code:
k x=srt(3,k)
---------------------
1 3.000000 =srt(3,1)
2 1.825455
4 1.563628
8 1.484080
16 1.457948
32 1.449171
64 1.446164
128 1.445135 =srt(3,128)
...
->inf -> ?? srt(3,inf)
================================
compare other limits
inf 1.444668 =e^(1/e)
--------------------------------
inf 1.442250 =3^(1/3)
On the other hand, it should arrive at 3^(1/3)...
Do I actually overlook something and the sequence can indeed cross e^(1/e)?
<urrks>
Gottfried
Gottfried Helms, Kassel