Summation of Divergent Power Series by Means of Factorial Series
Provides a numerical tool for summing divergent series in physics and mathematics, but the method is not new and the results are incremental.
Factorial series are shown to be effective for summing divergent power series, demonstrated by summing the asymptotic expansion for the exponential integral and the perturbation series for the quartic anharmonic oscillator.
Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are useful numerical tools for the summation of divergent (inverse) power series. This is documented by summing the divergent asymptotic expansion for the exponential integral $E_{1} (z)$ and the factorially divergent Rayleigh-Schrödinger perturbation expansion for the quartic anharmonic oscillator. Stirling numbers play a key role since they occur as coefficients in expansions of an inverse power in terms of inverse Pochhammer symbols and vice versa. It is shown that the relationships involving Stirling numbers are special cases of more general orthogonal and triangular transformations.