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# Summation of Series and Gaussian Quadratures

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

## Abstract

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.

## Keywords

Relative Error Weight Function Orthogonal Polynomial Gaussian Approximation Gaussian Quadrature
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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