Infinite Series

One typically first studies infinite series, in a standard calculus course, without going deeply into the theory of infinite series. In this chapter, using the results on the convergence of sequences, the authors prove the important theorems that concern an infinite series of real numbers. They introduce the concept of a convergent infinite series, and discuss a variety of different kinds of infinite series. The authors then examine a range of tests for convergence. The notion of convergence or divergence of an infinite series is defined in terms of the convergence or divergence of the sequence of its partial sums. The authors derive two new tests which often prove to be useful in showing that a series is absolutely convergent. These tests are called the Ratio Test and the Root Test. Finally, the authors investigate the effect of regrouping and rearranging the terms of an infinite series.