Accurate summation of the perturbation series for periodic eigenvalue problems

We show that algebraic approximants prove suitable for the summation of the perturbation series for the eigenvalues of periodic problems. Appropriate algebraic approximants: constructed from the perturbation series for a give n eigenvalue provide information about other eigenvalues connected with the chosen one by branch points in the complex plane. Such approximants also g ive those branch points with remarkable accuracy. We choose Mathieu's equat ion as illustrative example.