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+--- Thread: tiny q: superroots of real numbers x>e (/showthread.php?tid=227)
tiny q: superroots of real numbers x>e - Gottfried - 02/02/2009
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
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) = yCode:
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
RE: tiny q: superroots of real numbers x>e - bo198214 - 02/02/2009
Gottfried Wrote: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)?
Indeed a very interesting observation, Gottfried.
You only arrive at the expected value if it is \( <e \), i.e.
\( \lim_{k\to\infty} \text{srt}(x,k)=\sqrt[x]{x} \) only if \( 1\le x\le e \).
This is because \( 1\le {^\infty}b\le e \) for \( 1\le b\le e^{1/e} \), where \( x={^\infty}b \) and \( b=\sqrt[x]{x} \).
For \( x>e \), for example \( x=3 \), is always \( \text{srt}(x,k) > e^{1/e} \) for each \( k \). Suppose otherwise \( \text{srt}(x,k)\le e^{1/e}=:y \) then would \( x\le {^k}y \), for \( y\le e^{1/e} \) while \( {^k}y\le e \).
RE: tiny q: superroots of real numbers x>e - Gottfried - 02/03/2009
Hi Henryk -
It's late, I can't comment/proceed at the moment, let's see tomorrow.
Here are two plots to illustrate the beginning of the trajectory, anyway.
Nächtle... ;-)
[update] pic changed [/update]
Gottfried
RE: tiny q: superroots of real numbers x>e - bo198214 - 02/03/2009
To be clear: I think its sure that
\( \lim_{k\to\infty}\text{srt}(x,k) = \sqrt[x]{x} \) for \( 1\le x\le e \)
and for \( x>e \) I would guess:
\( \lim_{k\to\infty}\text{srt}(x,k) = e^{1/e} \)
this also corresponds to your pictures.
RE: tiny q: superroots of real numbers x>e - Gottfried - 02/03/2009
Yepp, so we have the interesting property, that we have two numbers:
a proper limit (e^(1/e)) for the sequence of srt of increasing order and x^(1/x) as value for "the immediate" evaluation of the infinite expression.
Hmm - surely this should be formulated more smoothly.
Can we then say, that the infinite iterative root for y>e has two values?
... so many questions...
Gottfried
RE: tiny q: superroots of real numbers x>e - bo198214 - 02/03/2009
Gottfried Wrote:Can we then say, that the infinite iterative root for y>e has two values?
No, we have two cases and for each case one limit.