On the Convergence of Iterated Exponentiation - I
Michael J. Creutz, R. M. Sternheimer · The Fibonacci Quarterly · 1980
We have investigated the properties of the function f(x) = x x * with an infinite number of # f s in the region 0 °o9 depending upon whether n is even or odd. An elementary exercise is to find a positive x satisfying (1) x x * ' = 2 when an infinite number of exponentiations is understood [1], [2], The standard solution is to note that the exponent of the first x must be 2, and thus x = /2. Indeed, the sequence fn defined by u-1 (2) f, = 2 J n+ 1 does converge to 2 as n goes to infinity. Now consider the problem (3) x*' " = |. By analogy9 one might assume that / n / 2