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Summation of series and Gaussian quadratures,II

Authors:	Gradimir V Milovanović

Institution:	(1) Faculty of Electronic Engineering, Department of Mathematics, University of Ni, P.O. Box 73, 18000 Ni, Serbia, Yugoslavia

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<v1 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- _m ² t ²))cos(z arctan( _m t, _m>=2/((2m+1)),m₀, involving the hyperbolic weightw(t)=1/cosh² t. Numerical results are included to illustrate the method.Dedicated to Luigi Gatteschi on the occasion of his 70th birthday

Keywords:	Primary 40A25 Secondary 30E20 65D32 33C45
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