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 reached the stage of enabling summation of the divergent tails of many series with an accuracy far beyond that of the smallest term. Several of these advances sprang from developments of Airy’s theory of waves near optical caustics such as the rainbow. Key understandings by Euler, Stokes, Dingle and Ecalle unify the different series corresponding to different parameter domains, culminating in the concept of resurgence: quantifying the way in which the low orders of such series reappear in the high orders.