Asymptotic summation of power series

Abstract

We give an asymptotic expansion in powers of n −1 of the remainder ∞ j=n f j z 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 , ν, a, b, p) = ∞ j=j 0 z j (j + b) ν−1 (j + a) −p , which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series.

A new summation method for power series with rational coefficients

“…Observe that the algorithm behavior gets worse in the neighborhood of z = 1 (i.e., for ω "small"), while it works satisfactorily at z = 1 (ω = 0); as already pointed out in [3], this could be ascribed to the discontinuity of the coefficients…”

“…where z ∈ C and f j ∈ C. In [3] we proved that, when the coefficients f j possess an asymptotic expansion f j ∼ a 1 j −p1 + a 2 j −p2 + ... , j → ∞ , 0 < p 1 < p 2 < ... , (2) and p k+1 − p k ∈ N, then also the remainder of (1) has an asymptotic expansion…”

“…Recall that, because of the asymptotic nature of the approximation in (15), the error term ε n,m tends to 0 as n → ∞, but is not infinitesimal as m → ∞; indeed, |ε n,m | exhibits for each fixed value of n a minimum in m which is infinitesimal as n → ∞. As in [3], however, we forgo a procedure based on finding (empirically) such a minimum for a given subsequence of indices {n s }: in fact, the complexity in the computation of σ n,m increases quadratically (z = 1) or cubically (z = 1) with m, so that this approach turns out to be quite costly.…”

“…[2], [3,Formula (24)]. Observe that the more general formula (5) of [3] contains a misprint: the number…”

“…An alternative strategy, which is that adopted in [3], consists in trying to balance the computational cost of the partial sum S n with that of σ n,m in terms of the order of magnitude of the respective flops number. While the computation of σ n,m requires basically only elementary arithmetic operations, that of S n may involve the evaluation of logarithms (when ν < 1).…”