Analysis of convergence of the 𝑇-transformation for power series

Abstract

Recently the present author has given some convergence theorems of general nature for Levin’s nonlinear sequence transformations. In this work these theorems are extended and sharpened to cover the case of power series, both inside and on their circle of convergence. It is shown that one of the two limiting processes considered in the previous work can be used for analytic continuation and a realistic estimate of its rate of convergence is given. Three illustrative examples are also appended.