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 $a_n=\frac{(-1)^{n-1}}{n}$ and $b_n=\frac{1}{n}$.
Alternating Series Test (AST): If the alternating series $\displaystyle\sum_{n=1}^\infty (-1)^{n-1}b_n=b_1-b_2+b_3-b_4+\cdots$ satisfies
then the series converges. |