Concerning the Class of a Group of Order

ing from the multiplication of 8\ by S2 . Special example : T* • T.* is absolutely convergent, but 1\ and T* are each divergent when r < § and * < §. In the divergent series 8/ the terms increase without limit in numerical value, as v increases without limit. The same is true of S2 . Herein lies the difference between this pair of divergent series yielding an absolutely convergent product, and the pair given by Pringsheim.* In the latter the terms of the divergent series remain finite as v increases indefinitely. From the relation S^SA ••• = 8l -82 p we see that there are cases in the multiplication of series in which divergent series may be used with safety—the sum of the final product series being convergent and equal to the product of the sums of the initially given convergent factor series, even when the product of some of the given factor series is divergent. If two or more convergent series, when multiplied together, yield a convergent product series s then the sum of this product series is equal to the product of the sums of the factor series. This theorem was proved by Abel for the case of two factor series,f and his method of proof is applicable to the general case. The extension is obvious.