Asymptotic summation of power series

We give an asymptotic expansion in powers of n-1 of the remainder ∑∝j=nfjzj, when the sequence fn has a similar expansion. Contrary to previous results, explicit formulas for the computation of the coefficients are presented. In the case of numerical series (z = 1), rigorous error estimates for the asymptotic approximations are also provided. We apply our results to the evaluation of S(z; j0, v, a, b, p) = ∑∞j=j0zj (j + b)v-1(j + a)-p, which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series.