Extrapolation methods and sequence transformations for computing slowly convergent integrals

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It is well known that in applied mathematics and in the numerical treatment of scientific and engineering problems, slowly convergent integrals and series occur very frequently. They are produced by approximation procedures depending on a parameter, perturbation techniques and reliable evaluation of functions that are defined by integrals. These slowly convergent integrals and series present severe numerical and computational difficulties. Traditional quadrature rules and summation techniques fail to provide accurate approximations to these integrals and series. An effective remedy for these problems is to use extrapolation methods and convergence acceleration. These methods have proved very useful for improving convergence of infinite series and infinite-range integrals but many challenges occur when dealing with complicated integrals.