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This review is focused on the borderline region of theoretical physics andmathematics. First, we describe numerical methods for the acceleration of theconvergence of series. These provide a useful toolbox for theoretical physicswhich has hitherto not received the attention it actually deserves. Theunifying concept for convergence acceleration methods is that in many cases,one can reach much faster convergence than by adding a particular series termby term. In some cases, it is even possible to use a divergent input series,together with a suitable sequence transformation, for the construction ofnumerical methods that can be applied to the calculation of special functions.This review both aims to provide some practical guidance as well as agroundwork for the study of specialized literature. As a second topic, wereview some recent developments in the field of Borel resummation, which isgenerally recognized as one of the most versatile methods for the summation offactorially divergent (perturbation) series. Here, the focus is on algorithmswhich make optimal use of all information contained in a finite set ofperturbative coefficients. The unifying concept for the various aspects of theBorel method investigated here is given by the singularities of the Boreltransform, which introduce ambiguities from a mathematical point of view andlead to different possible physical interpretations. The two most importantcases are: (i) the residues at the singularities correspond to the decay widthof a resonance, and (ii) the presence of the singularities indicates theexistence of nonperturbative contributions which cannot be accounted for on thebasis of a Borel resummation and require generalizations toward resurgentexpansions. Both of these cases are illustrated by examples.

From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions