  We show that the delta function potential can be exploited along with
perturbation theory to yield the result of certain infinite series. The idea is
that any exactly soluble potential if coupled with a delta function potential
remains exactly soluble. We use the strength of the delta function as an
expansion parameter and express the second-order energy shift as an infinite
sum in perturbation theory. The analytical solution is used to determine the
second-order energy shift and hence the sum of an infinite series. By an
appropriate choice of the unperturbed system, we can show the importance of the
continuum in the energy shift of bound states.
