Leçons sur les Séries Divergentes

Abstract

To make the object of this work intelligible, it is necessary to recall a few facts concerning infinite series in general. Suppose we have a sequence u₁, u₂, u₃, … u_n … (U) where u₁, u₂, &c., are analytical expressions constructed by a definite rule. Let s_n=u₁ + u₂ + … + u_n; then we have a derived analytical sequence s₁, s₂, s₃, … s_n, …; (S) this is a definite analytical entity, and its properties are implicitly fixed by those of the former sequence. The expressions u_n, s_n are, of course, functions of n; we may suppose, for simplicity, that, besides this, they involve, in addition to definite numerical constants, a single analytical variable, x. If we assign to x a numerical value, S becomes an arithmetical sequence, and three principal cases arise, according to the behaviour of s_n when n increases indefinitely. If s_n converges to a definite limit s we say that this is the sum of the series u₁ + u₂ + u₃ + …, and write s = u_n; but the ultimate value of s may be either indeterminate or infinite. In the second case Σu_n has no definite meaning; in the third we may say, if we like, that Σu_n is infinite, but this infinite sum is not a quantity with which we can operate, and presents no special interest.

Leçons sur les Séries Divergentes.

Par Émile Borel. Pp. viii + 184. (Paris: Gauthier-Villars, 1901.) Price fr. 4.50.