Summation of Series and Gaussian Quadratures

  • Gradimir V. Milovanović
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


In 1985, Gautschi and the author constructed Gaussian quadrature formulae on (0, +∞) involving Einstein and Fermi functions as weights and applied them to the summation of slowly convergent series which can be represented in terms of the derivative of a Laplace transform, or in terms of the Laplace transform itself. A problem that may arise in this procedure is the determination of the respective inverse Laplace transform. For the class of slowly convergent series, \(\sum\limits_{k = 1}^{ + \,\infty } {{{\left( { \pm 1} \right)}^k}} {a_k}with\,{a_k} = {k^{v - 1}}R\left( k \right) \)where 0 < v ≤ 1 and R(s) is a rational function, Gautschi recently solved this problem. In the present paper, using complex integration and constructing Gauss-Christoffel quadratures on (0, +∞) with respect to the weight functions w 1 (t) = 1/cosh2 t and w 2 (t) = sinh t/ cosh2 t, we reduce the series \(\sum\limits_{k\, = \,m}^{ + \,\infty } {f\left( k \right)} \,and\,{\sum\limits_{k\, = \,m}^{ + \,\,\infty } {\left( { - \,1} \right)} ^k}\,f\,\left( k \right)\) to weighted integrals of f involving weights w 1 and w 2, respectively. We illustrate this method with a few numerical examples.


Relative Error Weight Function Orthogonal Polynomial Gaussian Approximation Gaussian Quadrature 
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Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Gradimir V. Milovanović
    • 1
  1. 1.Faculty of Electronic Engineering, Department of MathematicsUniversity of NišNišYugoslavia

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