field now known as asymptotics beyond all orders or exponential asymptotics [@seg91]. One outcome of this relatively new field is that it seeks to obtain far more accurate values from the asymptotic expansions for functions/integrals than standard Poincare${\acute{\rm e}}$ asymptotics [@whi73]. These calculations, which often yield values that are accurate to more than twenty decimal places, are referred to as hyperasymptotic evaluations or hyperasymptotics, for short. Hence, exactification represents the extreme of hyperasymptotics.

In the first successful case of exactification exact values of a particular case of the generalized Euler-Jacobi series, viz. $S_3(a) =\sum_{n=1}^{\infty} \exp(-a n^3)$, were evaluated from its the complete asymptotic expansion, which was given in powers of $a$. Although it had been found earlier in Ref. [@kow95] that there could be more than one subdominant series in the complete asymptotic expansion for the generalized Euler-Jacobi series, the complete asymptotic expansion for $S_3(a)$ was found to be composed of an infinite dominant algebraic series and another infinite exponentially-decaying asymptotic series, whose coefficients resembled those appearing in the asymptotic series for the Airy function ${\rm Ai}(z)$. In carrying out the exactification of this complete asymptotic expansion, a range of values for $a$ was considered with the calculations performed to 