In this thesis we consider the convergence sets of formal power series of the form f(z, t)=sigma infinity j=0 pj(z)tj, where pj(z) are polynomials. A subset E of the complex plane C is said to be a convergence set if there is a series f(z, t)=sigma infinity j=0 pj(z)tj such that E is exactly the set of points z for which f(z, t) converges as a power series in t. A quasi-simply connected set is defined to be the union of a countable collection of polynomially convex compact sets. We prove that a subset of C is a convergence set if and only if it is a quasi-simply-connected set. We also give an example of a compact set which is not a convergence set.