  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.
