ASYMPTOTIC SUMMATION OF POWER-SERIES

We give an asymptotic expansion in powers of n(-1) of the remainder Si gma(j=n)(infinity) f(j)z(j), when the sequence f(n) has a similar expa nsion. Contrary to previous results, explicit formulas for the computa tion of the coefficients are presented. In the case of numerical serie s (z = 1), rigorous error estimates for the asymptotic approximations are also provided. We apply our results to the evaluation of S(z; j(0) , v, a: b,p) = Sigma(j=jo)(infinity) z(j) (j + b)(v-1)(j + a)(-p), whi ch generalizes various summation problems appeared in the recent liter ature on convergence acceleration of numerical and power series.