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It is very interesting reading Post Date: Sun, 27 Jul 2008 19:09:15 +0000 The most outspoken critic of the old methods in series was Abel. His letter to Ms friend Holmboe (1826) contains severe criticisms. It is very interesting reading, even to modern students. Autor of the post: Undefined Since we do not possess Post Date: Sun, 27 Jul 2008 18:53:35 +0000 In his demonstration of the binomial theo rem he established the theorem that if two series and their product series are all convergent, then the product series will converge towards the product of the sums of the two given series. This remarkable result would dispose of the whole problem of multiplication of series if we had a universal practical criterion of convergency for semi-convergent series. Since we do not possess such a criterion, theorems have been recently established by A. Autor of the post: Undefined Pringsheim reaches the following interesting Post Date: Sun, 27 Jul 2008 18:38:39 +0000 Pringsheim of Munich and A. Voss of Wiirzburg which remove in certain cases the necessity of applying tests of convergency to the product series by the application of tests to easier related expressions. Pringsheim reaches the following interesting conclusions: The product, of two semi-convergent series can never converge absolutely, but a semi-convergent series, or even a divergent series, multi plied by an absolutely convergent series, may yield an abso lutely convergent product. Autor of the post: Undefined Luckily, every one was found Post Date: Sun, 27 Jul 2008 18:24:33 +0000 The researches of Abel and Cauchy caused a considerable stir. We are told that after a scientific meeting in which Cauchy had presented his first researches on series, Laplace hastened home and remained there in seclusion until he had examined the series in Ms Moanique Cleste. Luckily, every one was found to be convergent! We must not conclude, however, that the new ideas at once displaced the old. Autor of the post: Undefined De Morgan established the loga Post Date: Sun, 27 Jul 2008 18:07:27 +0000 On the contrary, the new views were generally accepted only after a 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 Lud- wig Eaabe (Crelle, Vol IX); then follow those of De Morgan as given in his calculus. De Morgan established the loga rithmic criteria which were discovered in part independently by Bertrand. Autor of the post: Undefined It was the opinion Post Date: Sun, 27 Jul 2008 17:47:41 +0000 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 estab lishing logarithmic criteria. It was the opinion of Bonnet that the logarithmic criteria never fail ; but Du Bois-Rey- mond and Pringsheim have each discovered series demon- strably convergent in which these criteria fail to determine the convergence. Autor of the post: Undefined Among the first to suggest Post Date: Sun, 27 Jul 2008 17:34:32 +0000 The criteria thus far alluded to have been called by Fringsheim special criteria, because they all depend upon a comparison of the nfh. term of the series with special functions a n , n x , (logn), 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. Autor of the post: Undefined Dini of Pisa, Paul Du Post Date: Sun, 27 Jul 2008 17:17:47 +0000 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-Beyrnond, Gr. Autor of the post: Undefined Kummer s is a criterion Post Date: Sun, 27 Jul 2008 17:01:54 +0000 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 nth term, or the ratio of the (i + l)th term and the nth term, is made the basis of research. Kummer s is a criterion of the second kind. Autor of the post: Undefined The theory of Pringsheim Post Date: Sun, 27 Jul 2008 16:49:13 +0000 A criterion of the first kind, analogous to this, was invented by Pringsheim. From the general criteria established by Du Bois-Beyrnond and Prings heim 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 Mud, and also generalised criteria of the second kind, which apply, however, only to series with never increasing terms. 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