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 th term of the series with special functions , , , 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 th term, or the ratio of the th term and
the th term, is made the basis of research. Kummer's is a
criterion of the second kind. A criterion of the first kind, analogous to this, was invented by Pringsheim. From the general criteria established by Du Bois-Reymond and Pringsheim respectively, all the special criteria can be derived. The theory of Pringsheim is very complete, and offers, in addition
to the criteria of the first kind and second kind, entirely new criteria of a third kind, and also generalised criteria of the second kind, which apply, however, only to series with never
increasing terms. Those of the third kind rest mainly on the consideration of the limit of the difference either of consecutive terms or of their reciprocals. In the generalised criteria of the second kind he does not consider the ratio of two consecutive terms, but the ratio of any two terms however far apart, and deduces, among others, two criteria previously given by Kohn and Ermakoff respectively.