hyperasymptotic approach improving the accuracy of an asymptotic expansion when the asymptotic expansion is already very accurate. That is, often a relatively large value of the variable is chosen when the expansion has already been derived for large values. Although the hyperasymptotic approach may improve the situation for such large values, the method generally breaks down for small values, where standard Poincar${\acute{\rm e}}$ asymptotics also fails. In this work, however, the opposite will be done; exact values of a large variable expansion will be obtained for small and intermediate values of the variable, representing the regions where an asymptotic expansion is no longer valid. It should also be borne in mind that numbers do not lie, whereas “proving theorems" with the afore-mentioned symbols are unable to provide an indication as to just how accurate an expansion is and where it actually does break down.

Sec. 3 proceeds with a numerical investigation into the Borel-summed asymptotic forms given by Eq. (\[fiftyeight\]). In the first example the remainder is expressed as an infinite sum involving the incomplete gamma function, which means that Mathematica’s intrinsic routine is used to evaluate the remainder. To make contact with standard Poincar${\acute{\rm e}}$ asymptotics, initially a large 