*Proof of Corollary to Theorem 2*.

If the survival function *S*(*x*) is infinitely differentiable or *C*^{∞} at *x*_{0} but is not analytic, the Taylor expansion of *S*(*x*) about *x*_{0} converges to *S*(*x*) only at *x*_{0}. We assume to the contrary that there exists a Taylor expansion of *S*(*x*) about *x*_{0} 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 *x*_{0}:

Letting *C*_{p} = −*S*^{(p)}(*x*_{0}) and assuming there exists a smallest finite *p* ≥ 1 with *C*_{p} ≠ 0, then from Equation A.1,

Using the power series representation − log(1 − *x*) = *x* + *x*^{2}/2 + *x*^{3}/3 + … for | *x* | < 1, we obtain the following three expressions. If p>1,

(A.3)
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 ...