# How do you use the Alternating Series Test?

Oct 2, 2014

Alternating Series Test states that an alternating series of the form
${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n} {b}_{n}$, where ${b}_{n} \ge 0$,
converges if the following two conditions are satisfied:

1. ${b}_{n} \ge {b}_{n + 1}$ for all $n \ge N$, where $N$ is some natural number.
2. ${\lim}_{n \to \infty} {b}_{n} = 0$

Let us apply the test to the alternating series below.

${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n - 1} \frac{1}{\sqrt{n}}$

In this series, ${b}_{n} = \frac{1}{\sqrt{n}}$.

Let us check the two conditions.

1. $\frac{1}{\sqrt{n}} \ge \frac{1}{\sqrt{n + 1}}$ for all $n \ge 1$
2. ${\lim}_{n \to \infty} \frac{1}{n} = \frac{1}{\infty} = 0$

Hence, we conclude that the alternating series converges.

I hope that this was helpful.