Alternating Series

Introduction With the exceptions of geometric series, where $r$ may be negative, or the rare series with telescoping partial sums, the convergence tests we have worked with so far only work with positive-termed series. When the terms in a series can be positive or negative, things get more complicated; the sequence {$s_n$} of partial sums may not be monotonic, so it can be bounded yet divergent. This module will introduce the Alternating Series Test, which works on series in which the terms have alternating signs. Alternating Series and the Alternating Series Test An alternating series is a series $\displaystyle\sum_{n=1}^\infty a_n$ where $a_n$ has alternating signs. Notice that if $a_n$ has alternating signs, we will be able to let $b_n=\left\vert a_n \right\vert$, and write $a_n=(-1)^n b_n$ or $a_n=(-1)^{n-1}b_n$. For instance, $$\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac 14 + \ldots=\sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n}$$has terms...