Summation of divergent series: Order-dependent mapping

Summation methods play a very important role in quantum field theory because all perturbation series are divergent and the expansion parameter is not always small. A number of methods have been tried in this context, most notably Padé approximants, Borel–Padé summation, Borel transformation with mapping, which we briefly describe and one on which we concentrate here, Order-Dependent Mapping (ODM). We recall the basis of the method, for a class of series we give intuitive arguments to explain its convergence and illustrate its properties by several simple examples. Since the method was proposed, some rigorous convergence proofs were given. The method has also found a number of applications and we shall list a few.