Title: Divergent series: from Thomas Bayes’s bewilderment to today’s resurgence via the rainbow
Abstract: Following the discovery by Bayes in 1747 that Stirling’s series for the factorial is divergent, the study of asymptotic series has today culminated in ‘hyperasymptotics’: summing the divergent tails of many series with an accuracy far beyond the smallest term. Several of these advances sprang from Airy’s theory of waves decorating rainbows. Key understandings by Stokes, Dingle and Écalle unify the different series corresponding to different parameter domains. An application is to the histories of quantum transitions. The concept of resurgence quantifies how low orders of asymptotic series reappear in the high orders.