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Factorial series played a major role in Stirling's classic book "MethodusDifferentialis" (1730), but now only a few specialists still use them. Thisarticle wants to show that this neglect is unjustified, and that factorialseries are useful numerical tools for the summation of divergent (inverse)power series. This is documented by summing the divergent asymptotic expansionfor the exponential integral $E_{1} (z)$ and the factorially divergentRayleigh-Schr\"{o}dinger perturbation expansion for the quartic anharmonicoscillator. Stirling numbers play a key role since they occur as coefficientsin expansions of an inverse power in terms of inverse Pochhammer symbols andvice versa. It is shown that the relationships involving Stirling numbers arespecial cases of more general orthogonal and triangular transformations.

Summation of Divergent Power Series by Means of Factorial Series