# 44 analysis/tower.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 44 analysis/tower.p

R = N ^ (N ^ (N ^ ...)). What is the maximum N>0 that will yield a finite R?

analysis/tower.s

ANSWER: e^(1/e)

Let N be the number in question and R the result of the process. Then

R can be defined recursively by the equation:

(1) R = N^R

Taking the logarithm of both sides of (1):

(2) ln(R) = R * ln(N)

Dividing (2) by R and rearranging:

(3) ln(N) = ln(R) / R

Exponentiating (3):

(4) N = R^(1/R)

We wish to find the maximum value of N with respect to R. Find the

derivative of N with respect to R and set it equal to zero:

(5) d(N)/d(R) = (1 - ln(R)) / R^2 = 0

For finite values of R, (5) is satisfied by R = e. This is a maximum of

N if the second derivative of N at R = e is less than zero.

(6) d2(N)/d2(R) | R=e = (2 * ln(R) - 3) / R^3 | R=e = -1 / e^3 < 0

The solution therefore is (4) at R = e:

(7) Nmax = e^(1/e)

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