Convergence in divergent series related to perturbation methods using continued exponential and Shanks transformations

Abstract

Divergent solutions are ubiquitous with perturbation methods. We use continued function such as continued exponential to converge divergent series in perturbation approaches for energy eigenvalues of Helium, Stark effect and Zeeman effect on Hydrogen. We observe that convergence properties are obtained similar to that of the Pad\'e approximation which is extensively used in literature. Free parameters are not used which influence the convergence and only first few terms in the perturbation series are implemented.