Divergent series: from Thomas Bayes’s bewilderment to today’s resurgence via the rainbow

<!--HTML-->Following the discovery in 1747 that Stirling’s series for the factorial is divergent, the study of asymptotic series has advanced so that today it is possible to sum the divergent tails of many series with an accuracy far beyond that of the smallest term. Key advances by Euler, Stokes, Dingle and Écalle 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.