Riedel, Roland Asymptotische Entwicklung der Restsummen gewisser Potenzreihen. (German) Zbl 0571.41026 Wiss. Z., Martin-Luther-Univ. Halle-Wittenberg, Math.-Naturwiss. Reihe 33, No. 4, 109-120 (1984). In this carefully written paper, the author derives an asymptotic expansion (as \(n\to \infty)\) with recursively defined coefficients for the remainder after n terms in the series \(\sum^{\infty}_{v=2}z^ v v^{-\alpha}(\ell n v)^{-\beta}\) where z, \(\alpha\) and \(\beta\) are complex; the real case was considered in his earlier work [ibid. 32, No.3, 133-146 (1983; Zbl 0535.65001)]. The expansion is uniform in the set obtained from the region of convergence of the series by deleting a neighbourhood of 1. A suitable choice of n and the number of terms in the asymptotic expansion leads to a practical method of summing the series. The special case \(\alpha =1\), \(\beta =0\) is considered in detail and the results are compared with those arising from other methods of accelerating convergence including those due to R. Johnsonbaugh, Am. Math. Mon. 86, 637-648 (1979; Zbl 0425.65002) and A. C. Smith, Util. Math. 13, 249-269 (1978; Zbl 0388.65002). Reviewer: M.E.Muldoon MSC: 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 40A25 Approximation to limiting values (summation of series, etc.) 65B10 Numerical summation of series Keywords:power series; remainder sums; acceleration of convergence; correction terms; asymptotic expansion Citations:Zbl 0535.65001; Zbl 0425.65002; Zbl 0388.65002 PDFBibTeX XMLCite \textit{R. Riedel}, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-Naturwiss. Reihe 33, No. 4, 109--120 (1984; Zbl 0571.41026)