On Accelerating the Convergence of Infinite Double Series and Integrals

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]f£ 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 Sha...

Two New Classes of Nonlinear Transformations for Accelerating the Convergence of Infinite Integrals and Series

Two new classes of nonlinear transformations, the Dtransformation to accelerate the convergence of infinite integrals and the d-transformation to accelerate the convergence of infinite series, are presented. In the course of the development of these transformations two interesting asymptotic expansions, one for infinite integrals and the other for infinite series, are derived. The transformatio...

Infinite Series Forms of Double Integrals

where 1 2 1 2 , , , r r θ θ are any real numbers, and n is any positive integer. We can obtain the infinite series forms of these double integrals using Taylor series expansions and integration term by term theorem; these are the major results of this paper (i.e., Theorems 1-3). Adams et al. [1], Nyblom [2], and Oster [3] provided some techniques to solve the integral problems. Yu [4-29], Yu an...

The d2-transformation for infinite double series and the D2-transformation for infinite double integrals

New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate dand D-transformations. The D2transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the d2-transformation for double series works well for seri...

Uniform convergence of single and double sine series and integrals