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From: gjk@talcott.UUCP (Greg Kuperberg)
Newsgroups: net.math,net.puzzle
Subject: Re: x^x^x^... = 2
Message-ID: <427@talcott.UUCP>
Date: Thu, 2-May-85 01:16:35 EDT
Article-I.D.: talcott.427
Posted: Thu May  2 01:16:35 1985
Date-Received: Sat, 4-May-85 02:17:38 EDT
References: <505@petsd.UUCP>
Organization: Harvard University
Lines: 45
Xref: watmath net.math:1974 net.puzzle:814

> 		x ^ x ^ x ^ ... ^ x
> 		\________  _______/
> 		         \/
> 		      n times
> 
> tends to infinity with n if x = sqrt(2).  This seems to spring
> from associating the operators the "wring" way.  Messr Woods
> and Sonntag are parsing x ^ x ^ x as ((x ^ x) ^ x), whereas
> the "right" (i.e., conventional, conformist, wimpy) way to do
> it is as (x ^ (x ^ x)).

There's a reason to assume that the inner parentheses are on the right:
If they were on the left, then one could write the above expression
much more simply as x^(x^n).  Not only would there be a simple way of
stating the problem, the problem would be trivial:  If x>1 then x^(x^n)
clearly goes to infinity.

> 	Therefore, there is a limit of the sequence; call it
> y.  That is, x ^ y = 2.  The only possilbe value of y is 2.
> 	Problem: how fast does the sequence approach its
> limit?  That is, what is a rough estimate for 
> 
> 	2  -  x ^ x ^ ...^ x
> 
> with n x's, in terms of n?
...
> Full-Name:  Christopher J. Henrich

Let y(n)=x^y(n-1) and y(0)=1.  Eventually, y(n) will get fairly close to
2.  At this point we can approximate the function x^y by its derivative,
i.e. sqrt(2)^y is close to ln(2)*(y-2)+2, where ln() is the natural
logarithm.  From then on 2 - y(n) is a geometric sequence with ratio ln(2).
Thus 2-y(n) is close to C*ln(2)^n, where "C" is some constant which I don't
wish to find.

There is no doubt a rigorous proof of this, but I don't wish to find it, on
account of I'm lazy.  I guess I might as well at least ask the following
question:  What is the exact (as opposed to numerical) value of this
constant C which I mentioned above?
-- 
			Greg Kuperberg
		     harvard!talcott!gjk

"The eerily accurate drawing of Goetz showed the face of the 'before'
figure in comic-book ads for body-building devices."-Time Magazine, April 8