In this survey paper on modified approximants of continued fractions, the new idea is to regard pre-value regions as value regions for modified approximants. This is used to prove convergence and to derive truncation error estimates for such approximants. This point of view is now generally accepted and used.
Furthermore, the paper gives simple introductions to the methods of convergence acceleration and analytic continuation by means of modified approximants. The modifying factors are chosen by means of auxiliary continued fractions. This includes the “fixed point method” of Gill, Thron and Waadeland.