On the Convergence of Iterated Exponentiation - III
Michael J. Creutz, R. M. Sternheimer · The Fibonacci Quarterly · 1982
The present paper can be considered as an extension of two previous papers in which the properties of the following function were discussed (see [1] and [2]): (1) F(x9 y) = x^^ where an infinite number of exponentiations is understood. Equation (1) is the function specifically studied in [2]9 whereas in [1] we considered the simpler function (2) f(x) E F(x9 x)9 i.e., the case of Eq. (1) where x = y. For both Eqs. (1) and (2), the ordering of the exponentiations is important, and for Eq. (1) and throughout this paper, we mean a bracketing order "from the top down, " i.e., x raised to the power y9 followed by y raised to the power x^, and then x raised to the power y ^ xy \\ and so on, all the way down to the x which is at the lowest position of the "ladder." In the present paper, we study the properties of a function which is obtained by forming an infinite sequence of roots. We have restricted ourselves to a single (positive) variable x9 i.e., the analogue of Eq. (2). We will call this function $(x)9 and it is defined as follows: (3) (a?) = ' ^ / x 9-where an infinite number of roots is understood. The bracketing is again from "the top down, " i.e., we mean /x9 followed by the i/ST-th root of x9 which can be written as E,(x), followed by the root and so on, down to the lowest x in the "ladder." From Eq. (3), it can be see that we have: (4) $(x) = x* ^ = i/x9 provided that the sequence (3) has a nontrivial limit. From Eq. (4), we obtain the equation: (5) Hx) Hx) = x. Values of (j)(#) were calculated by means of a simple program embodying the sequential operations of Eq. (3) on a Hewlett-Packard calculator. In this manner, we have obtained the graph of Figure 1, in which $(x) is shown as a function of x. We note that for x (a0 has the value l/e = 0.36788. Indeed, for cj) = e " 1 9 Eq. (5) gives ^ » e-itt/'> = g-i/* = x, The reason for the abrupt decrease of $(x) to zero below x = e ~ 1,e is illustrated This paper was authored under contract EY-76-C-02-0016 with the U.S. Department