Convergent and Divergent Series in Physics

Understand

The aim of this review, based on a series of four lectures held at the 22nd "Saalburg" Summer School (2016), is to cover selected topics in the theory of perturbation series and their summation.

• The first part is devoted to strategies for accelerating the rate of convergence of convergent series, namely Richardson extrapolation, and the Shanks transformations, and also covers a few techniques for accelerating the convergence of Fourier series.
• The second part focuses on divergent series, and on the tools allowing one to retrieve information from them.
• These techniques include Euler summation, Borel summation, generic summation, and the method of continued functions, including in particular the Pad\'e theory based on continued fractions.