This work’s basic goal is to develop perturbation series expansions for the energy and the relevant wave function of a quantum system described by a screened Coulomb potential. The potentials having just a single screening parameter are considered in our studies. Then all entities, known and/or unknown, are expanded into Maclaurin series with respect to screening parameter. The main purpose is to develop recurrence relations amongst relevant entities. Since, the screening parameter power expansions havebeen considered, the relevant recursions become quite uncomplicated and analytic results can be obtained. Even though the first important issue is to obtain the perturbation components in energy and wave function of the system under consideration, the convergence issues are also very important. Depending on the state identity number, n, of the hydrogen atom limit states; the
perturbations series seem to be depending on not sole powers of γ but rather on the powers of (n + 1)2 γ. This may not be appearing so important at the first glance but may contract the convergence disks in the complex plane of γ. Another important convergence controlling agent seems to be the set of Uk Maclaurin coefficients of the screening function and more importantly their asymptotic behaviors at infinitely large k values. Depending on these behaviors the perturbation series’ convergence characteristics may change to give higher or lower evaluation capabilities through the perturbation series. The presentation will be including certain discussions on these aspects of the proposed perturbation approach and some practical applications will be given during the presentation to show the efficiency and possible pitfalls of proposed method.