The summation of power series and Fourier series

摘要

The well-known correspondence of a power series with a certain Stieltjes integral is exploited for summation of the series by numerical integration. The emphasis in this paper is thus on actual summation of series, rather than mere acceleration of convergence. It is assumed that the coefficients of the series are given analytically, and then the numerator of the integrand is determined by the aid of the inverse of the two-sided Laplace transform, while the denominator is standard (and known) for all power series.Since Fourier series can be expressed in terms of power series, the method is applicable also to them.The treatment is extended to divergent series, and a fair number of numerical examples are given, in order to illustrate various techniques for the numerical evaluation of the resulting integrals.