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Asymptotische Entwicklung der Restsummen gewisser Potenzreihen. (German) Zbl 0571.41026

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
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