Generalized Euler transformation for summing strongly divergent Rayleigh-Schrödinger perturbation series: The Zeeman effect

A generalized Euler transformation (GET) is introduced which provides a powerful alternative method of accurately summing strongly divergent Rayleigh-Schrödinger (RS) perturbation series when other summability methods fail or are difficult to apply. The GET is simple to implement and, unlike a number of other summation procedures, requires no a priori knowledge of the analytic properties of the function underlying the RS series. Application of the GET to the difficult problem of the RS weak-field ground-state eigenvalue series of the hydrogen atom in a magnetic field (quadratic Zeeman effect) yields sums of good accuracy over a very wide range of field strengths up to the most intense fields of ${10}^{14}$ G. The GET results are compared with those obtained by other summing methods.