On accelerating the convergence of infinite double series and integrals

The generalization of Shanks’ e-transformation to double series is discussed and a class of nonlinear transformations, the ${[A/S]_R}$ transformations, for accelerating the convergence of infinite double series is presented. It is constructed so as to sum exactly infinite double series whose terms satisfy certain finite linear double difference equations; in that sense it is a generalization of Shanks’ e-transformation or its equivalent Wynn’s $\varepsilon$-algorithm. A generalization of the ${[A/S]_R}$ transformation to N-dimensional series is also presented and their application to power series is discussed and exemplified. Some transformations for accelerating the convergence of infinite double integrals are also obtained, generalizing the confluent $\varepsilon$-algorithm of Wynn and the G-transformation of Gray, Atchison, and McWilliams for infinite 1-D integrals.