On the stability of a class of convergence acceleration methods for power series

Abstract

in this paper we study the problem of evaluating the sum of a power series whose terms are given numerically with a moderate accuracy. For a large class of divergent series a sum may be defined using analytic continuation. This sum may be estimated using the values of a finite number of terms. However, it is established here that the accuracy of this estimate will generally deteriorate if we use an ever-growing number of terms. A result on the stability of product quadrature is also obtained as a corollary of our main stability theorem.

Stable convergence acceleration using Laplace transforms

“…Hence it should be possible to evaluate f(z) from (1.1) outside the region of convergence as well. In [7] Hence we can easily construct g, if a, is a polynomial in -to the case when a, has a power expansion in (n+l)~ -1 is straightforward. We mention here the two special cases f(1)= ~ a., f(-l)= ~ (-l)"a., where g is a known function satisfying the constraint ~ e-'lg(t)l dt < ~.…”