The summation of power series and Fourier series (Q1063374)

For evaluating a sum of a power series \(s(x)=u_ 1-u_ 2x+u_ 3x^ 2- ... \), of a variable x, the coefficients \(u_ k\) are revealed as the moments \(u_ k=\int^{\infty}_{0}u^{k-1}f(u)du\) of the function f(u). In this paper, the f(u) is determined by the inversion of the two- sided Laplace transform of the \(u_ k\). Thus, the summation results in numerical calculation of the Stieltjes integral \(I(x)=\int^{\infty}_{0}f(u)/(1+ux)du.\)