Page:A History of Mathematics (1893).djvu/356

 severe and long struggle. As late as 1844 De Morgan began a paper on "divergent series" in this style: "I believe it will be generally admitted that the heading of this paper describes the only subject yet remaining, of an elementary character, on which a serious schism exists among mathematicians as to the absolute correctness or incorrectness of results."

First in time in the evolution of more delicate criteria of convergence and divergence come the researches of Josef Ludwig Raabe (Crelle, Vol. IX.); then follow those of De Morgan as given in his calculus. De Morgan established the logarithmic criteria which were discovered in part independently by J. Bertrand. The forms of these criteria, as given by Bertrand and by Ossian Bonnet, are more convenient than De Morgan's. It appears from Abel's posthumous papers that he had anticipated the above-named writers in establishing logarithmic criteria. It was the opinion of Bonnet that the logarithmic criteria never fail; but Du Bois-Reymond and Pringsheim have each discovered series demonstrably convergent in which these criteria fail to determine the convergence. The criteria thus far alluded to have been called by Pringsheim special criteria, because they all depend upon a comparison of the $\scriptstyle{n}$th term of the series with special functions $\scriptstyle{a^n}$, $\scriptstyle{n^n}$, $\scriptstyle{n(\log n)^n}$, etc. Among the first to suggest general criteria, and to consider the subject from a still wider point of view, culminating in a regular mathematical theory, was Kummer. He established a theorem yielding a test consisting of two parts, the first part of which was afterwards found to be superfluous. The study of general criteria was continued by U. Dini of Pisa, Paul Du Bois-Reymond, G. Kohn of Minden, and Pringsheim. Du Bois-Reymond divides criteria into two classes: criteria of the first kind and criteria of the second kind, according as the general $\scriptstyle{n}$th term, or the ratio of the $\scriptstyle{(n+1)}$th term and