Asymptotic summation of power series

Summary.

We give an asymptotic expansion in powers of $n^{-1}$ of the remainder $\sum_{j=n}^\infty f_jz^j$, when the sequence $f_n$ 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 $S(z;j_0,\nu,a,b,p)=\sum_{j=j_0}^\infty z^j$ $(j+b)^{\nu-1}(j+a)^{-p}$, which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series.