Certain Investigations in the Summability of Series

• Main Author: Pandey, Amaresh Kumar
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2005
• Subjects: Mathematics
• ISBN: 9780355902310; 0355902311

The old hazy notion of convergence of infinite series was placed on sound foundation with the appearance of Cauchy’s monumental work “coursed’ Analyses algebrique” in 1821 and Abel’s researches on the Binomial series in 1826. However, it was observed that there were certain non-convergent series which particularly in Dynamical Astronomy, furnished nearly correct results. A theory of divergent series was formulated explicitly for the first time in 1890, when Cesáro published a chapter on the multiplication of series. Since then the theory of series, whose sequence of partial sums oscillates, has been the center of attraction and fascination for most of the pioneering mathematical analysts. After persistent efforts in which a number of celebrated mathematicians took part, it was only in the closing decade of the lost century and in the early years of the present century that satisfactory methods were devised so as to associate with them by processes closely connected with Cauchy’s concept of convergence, certain values which may be called their “sums” in a reasonable way. These processes of associating generalized sums known as methods of summability. (Szász and Hardy) provide a natural generalization of the classical notion of convergence (Hobson) and are thus responsible for bringing within the applicability a wider class of erstwhile rejected series that used to be tabooed as divergent.