VII. A theory of asymptotic series

Abstract

Abstract In their efforts to place mathematical analysis on the firm est possible foundations, Abel and Cauchy found it necessary to banish non-convergent series from their work ; from that time until a quarter of a century ago the theory o f divergent series was, in general, neglected by mathematicians. A consistent theory of divergent series was, however, developed by Poincaré in 1886, and, ten years later, Borel enunciated his theory of summability in connection with oscillating series. So far as diverging power series are concerned, the theory of Borel is more precise than that of Poincaré.