Rayleigh-Schrödinger perturbation theory for a hydrogen atom in an electric field and for a harmonic oscillator with anharmonicity gx4 are two examples of divergent power series – whose simplest prototype is the factorially divergent series 1-z+2!z2-3!z3+4!z4-5!z5+… . The main computational tool to sum such divergent series is the Borel-Padé method. We present a new summation method that is much more powerful than Borel-Padé, but which includes Borel-Padé as a special case: the sum of a divergent series as a linear combination of “previously summed” divergent series. The linear coefficients and “previously summed” series are chosen to “match” the coefficients of the series to be summed. The choice of which “previously summed” series is where “educated” comes in. The educated-match algorithm leans heavily on the theory of Padé approximants, is easy to use, and in some examples yields dramatically accurate numerical results over a wide range from only a few terms in the series.

Summation by educated match: A new, powerful technique to sum divergent series

255th National Meeting (2018)

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