On the Convergence of Borel Approximants

Many kinds of so-called ``irregular singular'' problems arising from differential equations have formal power series solutions that are everywhere divergent. These series, however, are asymptotic expansions for actual solutions of the equations and we consider a method (Borel summation) to obtain convergent representations for these functions. This can be thought of as a change of basis from power series to other functions and we look for the rate of convergence and the convergence domains. A general theorem will be presented and then several types of applications will be discussed.