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From: cjh@petsd.UUCP (Chris Henrich)
Newsgroups: net.math,net.puzzle
Subject: x^x^x^... = 2
Message-ID: <505@petsd.UUCP>
Date: Tue, 30-Apr-85 19:22:25 EDT
Article-I.D.: petsd.505
Posted: Tue Apr 30 19:22:25 1985
Date-Received: Wed, 1-May-85 04:32:47 EDT
Organization: Perkin-Elmer DSG, Tinton Falls, N.J.
Lines: 39
Xref: watmath net.math:1967 net.puzzle:804

[]
	John Woods and Jeff Sonntag have posted articles about
this problem, containing the statement that 

		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)).
	With the latter method, if x = sqrt(2), then the
value of the iterated exponentiation must be less than 2.
Proof is by induction: if its true for n-1 operands, then you
can prove it for n. (Try it, it is not very hard.)  And it is
true for one operand, since x < 2.
	On the other hand, the sequence of values does
increase as n increases.  Again, it's easy to prove
inductively.
	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?

Regards,
Chris

--
Full-Name:  Christopher J. Henrich
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