Ebook: The Art of the Infinite: The Pleasures of Mathematics

1. On page 98, the authors mis-define what they call a tower, defining it as x raised to the x-th power, then that raised to the x-th power, and so on and so on. In other words, if f(n) is defined to be the value of the tower after n steps, then:

f(n+1) = f(n)^x, for every positive integer n.

However, with that definition, and x= sqrt(2), the tower's terms quickly diverge - A spread sheet shows that f(20) is greater than 10 to the 108-th power. Instead, they should have defined the tower by the rule:

g(n+1) = x^g(n), for every positive integer n.

With that definition, and x = sqrt(2) the series {g(n)} converges to the number 2, as they claim. Indeed, a spread sheet shows that g(20) is approximately 1.999586. Also, with this new definition, the book's proof is legitimate, because the proof used the rule:

x^y = y, where y is the limiting value of the tower as n approaches infinity. That equation is not true with the book's definition of y, because it uses f(n) instead of g(n).

The heart of the matter is that the operation of taking successive powers is not "associative", even when the sequence is finite. For example,

(3^3)^3 = 27^3 = 19683, and that is not equal to

3^(3^3) = 3^27 = 7.6256E+12. Associativity fails!

2. The first term in the equation at the top of page 97 should be 1 - 1/2,
not (1 - 1/2)/2.

George Monser