Intended for advanced undergraduates and graduate students, this concise text focuses on the convergence of real series. Definitions of the terms and summaries of those results in analysis that are of special importance in the theory of series are specified at the outset. In the interests of maintaining a succinct presentation, discussion of the question of the upper and lower limits of a function is confined to an outline of those properties with a direct bearing on the convergence of series. The central subject of this text is the convergence of real series, but series with complex terms and real infinite products are also examined as illustrations of the main theme. Infinite integrals appear only in connection with the integral test for convergence. Topics include functions and limits, real sequences and series, series of non-negative terms, general series, series of functions, the multiplication of series, infinite products, and double series. Prerequisites include a familiarity with the principles of elementary analysis.

This text for advanced undergraduate and graduate students presents a rigorous approach that also emphasizes applications. Encompassing more than the usual amount of material on the problems of computation with series, the treatment offers many applications, including those related to the theory of special functions. Numerous problems appear throughout the book. The first chapter introduces the elementary theory of infinite series, followed by a relatively complete exposition of the basic properties of Taylor series and Fourier series. Additional subjects include series of functions and the applications of uniform convergence; double series, changes in the order of summation, and summability; power series and real analytic functions; and additional topics in Fourier series. The text concludes with an appendix containing material on set and sequence operations and continuous functions.