Series: Convergence and Divergence

The main goal of this chapter is to examine the theory and applications of infinite sums, which are known as infinite series . In Section 5.1, we introduce the concept of convergent infinite series, and discuss geometric series, which are among the simplest infinite series. We also discuss general properties of convergent infinite series and applications of geometric series. In Section 5.2, we examine various tests for convergence so that we can determine whether a given series converges or diverges without evaluating the limit of its partial sums. Our particular emphasis will be on divergence tests, and series of nonnegative numbers, and harmonic p -series. In Section 5.3, we deal with series that contain both positive and negative terms and discuss the problem of determining when such a series is convergent. In addition, we look at what can happen if we rearrange the terms of such a convergent series. We ask, Does the new series obtained by rearrangement still converge ? A remarkable result of Riemann on conditionally convergent series answers this question in a more general form. Finally, we also deal with Dirichlet’s test and a number of consequences of it. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.