New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field

The perturbation series for the ground state energy of the hydrogen atom in the external magnetic field is summed via the sequence transformations. The formula for the large-order behavior of the partial sums of the series is derived. From this formula a new general sequence transformation is suggested. This transformation contains free parameters that can be further optimized. It is shown that if the renormalization approach is used, the optimal choice of these parameters leads to the previously suggested Weniger transformation