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.