Ideas from continued fraction theory extended to Pade approximation and generalized iteration

This is a survey of some basic ideas in the convergence theory for continued fractions, in particular value sets, general convergence and the use of modified approximants to obtain convergence acceleration and analytic continuation. The purpose is to show how these ideas apply to some other areas of mathematics. In particular, we introduce {w(k)}-modifications and general convergence for sequences of Pade approximants.