Proof of Corollary to Theorem 2.
If the survival function S(x) is infinitely differentiable or C∞ at x0 but is not analytic, the Taylor expansion of S(x) about x0 converges to S(x) only at x0. We assume to the contrary that there exists a Taylor expansion of S(x) about x0 with a positive radius of convergence. To study the limiting behavior of the expression in the left-hand side of Equation 2.32, we express S(x) as the Taylor series about x0:
Letting Cp = −S(p)(x0) and assuming there exists a smallest finite p ≥ 1 with Cp ≠ 0, then from Equation A.1,
Using the power series representation − log(1 − x) = x + x2/2 + x3/3 + … for | x | < 1, we obtain the following three expressions. If p>1,
The expressions in Equations A.2, A.4, and A.5 allow us to re-express the limit in the left-hand side of Equation 2.32 as follows:
If p=1, Equation A.6 is easily seen to equal 0 directly. This shows ...