Asymptotic series, divergent series, and Tao's method: the Casimir effect

Using certain series which occur in quantum electrodynamics (qed) as a prototype, we provide a pedagogic review of asymptotic and divergent series and some of their applications in physics. We first review the concept of asymptotic series, and a few of its associated deep conceptual problems in qed. We then discuss summability methods for divergent series and their limitations. In the core of the paper, it is shown that Tao's recent method of smoothed sums, which conveys a precise mathematical sense to the "sums of the series" obtained by some of these summability methods, also provides a rigorous mathematical theory of an important effect in non-perturbative qed, the Casimir effect for perfectly conducting parallel plates.