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Divergent series: taming the tails

By the 17th century, in what became the theory of convergent series, it was beginning to be understood how a sum of infinitely many terms could be finite; this is now a fully developed and largely standard element of every mathematician’s education. Contrasting with it is the theory of series that do not converge, especially those in which the terms first get smaller but then increase factorially: this is the class of ‘asymptotic series’, encountered frequently in applications, with which this article is mainly concerned. Although now a vibrant area of research, the development of the theory of divergent series has been tortuous and often accompanied by controversy. As a pedagogical device to explain the subtle concepts involved, we will focus on the contributions of individuals and describe how the ideas developed during several (overlapping) historical epochs, often driven by applications ranging from wave physics to number theory. This article complements the accompanying Companion article by P D Miller on ‘Perturbation Theory (including Asymptotic Expansions)’.