prove anything in standard Poincar${\acute{\rm e}}$ asymptotics, especially when an asymptotic expansion becomes divergent or possesses an infinity in its remainder. As a consequence, asymptotics is often ridiculed by pure mathematicians for being vague and limited in range of applicability. E.g., when an expansion is described as being valid for large and small values of a variable, invariably one does not know the actual value when this applies or indeed, just how accurate it is compared with the original function. These symbols, however, are not employed here at any stage, having become redundant due to the concept of regularization. Nevertheless, the reader may still feel uncomfortable or uncertain about the results given in Sec. 2. Therefore, Sec. 3 presents an extensive numerical investigation aimed at verifying the results in Theorem 2.1. As will be observed, since the results from these studies are exact, they are far more accurate than the alternative hyperasymptotic approach of developing strategies for truncating asymptotic series beyond the optimal point of truncation as discussed in Refs.  [@ber90]-[@ber91a]. In addition, numerical studies can reveal many interesting properties, provided they are carried out in an appropriate manner. This refers to situations where claims are often made about a 