A series in which the terms are alternately positive and negative is an [i]alternating series[/i]. For example, here is the [i]alternating harmonic series[/i]:[br] [math]1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots+\frac{\left(-1\right)^{n+1}}{n}+\cdots[/math] [b][br]The Alternating Series Test[/b]: The series[br] [math]\sum_{n=1}^{\infty}\left(-1\right)^{n+1}u_n=u_1-u_2+u_3-u_4+\cdots[/math][br]converges if all three of the following conditions are satisfied:[list=1][*]The [math]u_n[/math]'s are all positive.[/*][*]The positive [math]u_n[/math]'s are (eventually) nonincreasing: [math]u_n\ge u_{n+1}[/math] for all [math]n\ge N[/math], for some integer [math]N[/math].[/*][*][math]u_n\to0[/math].[/*][/list]

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]