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Summation of series and Gaussian quadratures, II

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Abstract

Continuing previous work, we discuss applications of our summation/integration procedure to some classes of complex slowly convergent series. Especially, we consider the series of the form\(\sum\nolimits_{k = 1}^{ + \infty } {( \pm 1)^k k^{v - 1} } R(k)\), where 0<v≦1 andR(s) is a rational function. Such cases were recently studied by Gautschi, using the Laplace transform method. Also, we give an appropriate method for calculating values of the Riemann zeta function\(\zeta (z) = \sum\nolimits_{k = 1}^{ + \infty } {k^{ - z} } \), which can be transformed to a weighted integral on (0,+∞)of the functiont → exp (−z/2)log(1-β 2 m t 2))cos(z arctan(β m t,β m >=2/((2m+1)π),m∈ℕ0, involving the hyperbolic weightw(t)=1/cosh2 t. Numerical results are included to illustrate the method.

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Authors and Affiliations

  1. Faculty of Electronic Engineering, Department of Mathematics, University of Niš, P.O. Box 73, 18000, Niš, Serbia, Yugoslavia

    Gradimir V. Milovanović

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  1. Gradimir V. Milovanović
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Dedicated to Luigi Gatteschi on the occasion of his 70th birthday

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Milovanović, G.V. Summation of series and Gaussian quadratures, II. Numer Algor 10, 127–136 (1995). https://doi.org/10.1007/BF02198299

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