We prove that certain universality properties of the partial sums force a power series to have Ostrowski-gaps. This has interesting consequences for some classes of universal functions.

Let (P n ) be a sequence of polynomials which converges with a geometric rate on some arc in the complex plane to an analytic function. It is shown that if the sequence has restricted growth on a closed plane set E which is non-thin at ∞, then the limit function has a maximal domain of existence, and (P n ) converges with a locally geometric rate on this domain. If (s n k ) is a sequence of partial sums of a power series, a similar growth restriction on E forces the power series to have Ostrowski gaps. Moreover, the requirement of non-thinness of E at ∞ is necessary for these conclusions.

In this paper, series of rational functions with fixed poles, which have restricted growth near the poles are considered. If they converge with a geometric rate on a continuum, a phenomenon of overconvergence takes place, in the sense that the convergence extends to a certain maximal domain. From this result, some properties of universal Laurent series are derived.