The field of infinite series is so large that any investigation into that field must necessarily be limited to a particular phase. An attempt has been made to develop a number of tests having a wide range of applications. Particular emphasis has been placed on tests for series of positive terms.

The purpose of this paper is to examine certain questions concerning infinite series. The first chapter introduces several basic definitions and theorems from calculus. In particular, this chapter contains the proofs for various convergence tests for series of real numbers. The second chapter deals primarily with the equivalence of absolute convergence, unconditional convergence, bounded multiplier convergence, and c0 multiplier convergence for series of real numbers. Also included in this chapter is a proof that an unconditionally convergent series may be rearranged so that it converges to any real number desired. The third chapter contains a proof of the Silverman-Toeplitz Theorem together with several applications.

This thesis attempts to establish properties of Hölder and Cesàro summable series analogous to those of ordinary convergent series and also to establish properties that are possibly different from those of convergent series.

This thesis is conceived as a learning text, which deals with infinite numerical series. The thesis thematicaly results from curriculum for lectures of fourth therm's analysis for field called "Pedagogy of mathematics for secondary schools" on Pedagogical faculty by University of South Bohemia. It contains theoretical interpretation of basic concepts and theorems about infinite series, including of proofs of shown theorems, plenty of solved excersises, which show aplication of these concepts and theorems, and collection of excercises to study indipendently as well.

In this thesis the author presents a history of non-convergent series which, in the past, played an important role in mathematics. Euler's formula, Stirling's series and Poincare's theory are examined to show the development of asymptotic series, a subdivision of divergent series.

This paper presents new concepts in the use of infinite and infinitesimal numbers in real analysis. theory is based upon the hyperreal number system developed by Abraham Robinson in the 1960's in his invention of "nonstandard analysis". paper begins with a short exposition of the construction of the hyperreal nU1l1ber system and the fundamental results of nonstandard analysis which are used throughout the paper. The new theory which is built upon this foundation organizes the set hyperreal numbers through structures which on an infinite base logarithm. Several new relations are introduced whose properties enable the simplification of calculations involving infinite and infinitesimal The paper explores two areas of application of these results to standard problems in elementary calculus. The first is to the evaluation of limits which assume indeterminate forms. The second is to the determination of convergence of infinite series. Both applications provide methods which greatly reduce the amount of con1putation necessary in many situations. / Master of Science

Submitted by VALERIA OSTETE JANNIS LUCHETTA null (v_luchetta@uol.com.br) on 2017-12-18T19:08:57Z No. of bitstreams: 1 Tese-Valeria_Ostete_Jannis_Luchetta.pdf: 50620362 bytes, checksum: 83603c2cf7954c3e54c5b514f1349a73 (MD5) / Approved for entry into archive by Adriana Aparecida Puerta null (dripuerta@rc.unesp.br) on 2017-12-19T16:13:23Z (GMT) No. of bitstreams: 1 luchetta_voj_dr_rcla.pdf: 50476781 bytes, checksum: b699e31dcd81595fcb782249edc5c527 (MD5) / Made available in DSpace on 2017-12-19T16:13:23Z (GMT). No. of bitstreams: 1 luchetta_voj_dr_rcla.pdf: 50476781 bytes, checksum: b699e31dcd81595fcb782249edc5c527 (MD5) Previous issue date: 2017-11-24 / No presente trabalho apresentamos uma análise de alguns dos capítulos da obra Elements of Algebra (1840), de Leonhard Euler (1707 - 1783), que tratam de Séries infinitas. Nesta obra encontramos os métodos e os resultados mais importantes à respeito de álgebra alcançados por Euler até 1770. Nosso objetivo foi analisar e evidenciar os diferentes modos de produção de significados e conhecimentos para o objeto matemático séries infinitas na obra supra citada tomando como fundamentação teórica e metodológica o Modelo dos Campos Semânticos. Apresentamos a tradução dos capítulos selecionados, produzimos significados a eles utilizando nosso referencial teórico e os comparamos com a forma que produzimos significados e conhecimentos hoje utilizando a Teoria de Séries. / In this work we present an analysis of some of the chapters of Leonhard Euler’s (1707- 1783) Elements of Algebra (1840), which deal with Infinite Series. In his work we find the most important methods and results regarding algebra achieved by Euler until 1770. Our goal was to analyze and evidence the different modes of production of meanings and knowledge for the mathematical object infinite series in the work cited above taking as theoretical and methodological foundation the Model of Semantic Fields. We present the translation of the selected chapters, we produce meanings for them using our theoretical benchmark and compare them with the way we produce meanings and knowledge today using the Theory of Series.