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We give an apparently new proof of Stirling's original asymptotic formula forthe behavior of $\ln z!$ for large $z$. Stirling's original formula is not theformula widely known as "Stirling's formula", which was actually due to DeMoivre. We also show by experiment that this old formula is quite effective fornumerical evaluation of $\ln z!$ over $\mathbb{C}$, when coupled with thesequence acceleration method known as Levin's $u$-transform. As an homage toStirling, who apparently used inverse symbolic computation to identify theconstant term in his formula, we do the same in our proof.

Stirling's Original Asymptotic Series from a Formula like one of Binet's and its Evaluation by Sequence Acceleration