On the Determination of the Intermediate Point in Taylor's Theorem

Abstract

But no additional information is given about the location of ( within (a, x) (suppose momentarily a < x). S. Haber and 0. Shisha [3] showed that under suitable conditions the point f lies in the left half of (a, x). Here a method is shown of approximating f by means of a sequence converging to it. By Rolle's Theorem it is a simple matter to see that if f(n+ 1) exists and does not vanish in I then the point 5 that solves (1) is unique; in this case, f is a well defined single valued function of x, 6 = 6(x). We make the convention ((a) = a so that the function 6 is continuous in I. Let n be a positive integer, which will be fixed throughout this article. Given a function f as above, we shall denote F(f) the set {k 2 1: f(n+k)(a) + 0}. Suppose that F(f) is not empty for some f. The minimum of SF(f) will be denoted by v. Finally, A will be the number defined by A = (n n v) /.