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