Divergent series and asymptotic expansions, 1850-1900

Divergent series enjoyed a period of free and easy use in the eighteenth century. This was somewhat curtailed in the early nineteenth century even though interesting applied problems were solved using asymptotic expansions of divergent series during that time. In the second half of the nineteenth century asymptotic expansions were placed on a firm and rigorous foundation such that their use become standard practice by early in the twentieth century. I elucidate how and why that happened by examining the work on asymptotic expansions of George Gabriel Stokes, Jules Henri Poincaré and Thomas Jan Stieltjes.