AC Alternating Series

Section 8.4 Alternating SeriesMotivating QuestionsWhat is an alternating series?Under what conditions does an alternating series converge? Why?How well does the \(n\)th partial sum of a convergent alternating series approximate the actual sum of the series? Why?So far, we’ve considered series with exclusively nonnegative terms. Next, we consider series that have some negative terms. For instance, the geometric series\begin{equation*}2 - \frac{4}{3} + \frac{8}{9} - \cdots + 2 \left(-\frac{2}{3} \right)^n + \cdots\text{,}\end{equation*}has \(a = 2\) and \(r = -\frac{2}{3}\text{,}\) so that every other term alternates in sign. This series converges to\begin{equation*}S = \frac{a}{1-r} = \frac{2}{1- \left(-\frac{2}{3}\right)} = \frac{6}{5}\text{.}\end{equation*} In Preview Activity8.4.1 and our following discussion, we investigate the behavior of similar series where consecutive terms have opposite signs. Preview Activity 8.4.1.Preview Activity8.3.1 showed how we can approximate the number...