Re: tetration - inf superroot
- From: "alainverghote@xxxxxxxxx" <alainverghote@xxxxxxxxx>
- Date: Tue, 3 Feb 2009 02:52:24 -0800 (PST)
On 2 fév, 20:55, "Ioannis" <morph...@xxxxxxxxxxxx> wrote:
Gottfried Helms wrote:
[snip]
Hi Ionannis -
nice to read from you!
But I can't follow one argument:
As you correctly found, the function:
f(x) = x^(1/x) is a partial inverse of the power tower.
The final value of the iteration depends on the value of x.
Because the function f(x) has a maximum at x = e, this value (i.e.,
~1.444667861) will be attained ONLY when the the power tower has as base x =
e.
Any other iteration for rit(x,k) will converge to the value given by f(x).
Plot the function f(x) from 0 to 5 say, and see the graph. For x = 3, the
iteration will converge to
f(3) ~= 1.442249570
But this would mean, that the base
x_k = 1.44466... (=e^(1/e) = ß)
must be crossed while "k is not yet infinite", sloppily said, if we assume,
No. That's where the mistake lies. The function f(x) = x^(1/x) is a partial
inverse of the *infinite powertower* x^x^....
Hence, you won't ever get the result you are looking for, for any _finite_ k :-)
You can get only approximations, when k is finite.
Try it the other way around: Choose a value x in [(1/e)^e, e^(1/e)].
Calculate y = x^(1/x). Then,
lim_{n->oo} y^y^... = x.
For any finite iteration, you won't be able to reach x.
that x_1 = 3,
x_2 = 1.825
x_3 =
... decreasing for increasing k
x_k = 1.44466... = ß // "for some finite k"
!? ...
and in the
limit x_inf = 1.4422... (=3^(1/3)) (for k->inf)
or, as you say:
the iteration to converge to something less than e^(1/e), thereby crossing
e^(1/e) from above, but settling on the corresponding value given by f(x),
for
But for which *finite* k will then the inverse function f(ß,k) = 3 ?
Or do we assume that the trajectory of x_k misses ß and goes
through the complex plane?
[snip]
Gottfried
--
Ioannis- Masquer le texte des messages précédents -
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Dear Friends,
Just two things :
1°)There is a known 'window of convergence' for infinite
towers :x in [(1/e)^e , e^(1/e)] or [0.659..,1.444667861] (1)
2°)In this case the real limit is equal to
L = -W(-ln(x))/ln(x)
Fixed points verify |x^L*ln(x)|=1
namely x inside interval (1)
Have a nice day,
Alain
.
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