# How do you use the Nth term test for divergence on an alternating series?

Nov 26, 2014

If an alternating series has the form

${\sum}_{n = 0}^{\infty} {\left(- 1\right)}^{n} {b}_{n}$,

where ${b}_{n} \ge 0$,

then the series diverges if

${\lim}_{n \to \infty} {b}_{n} \ne 0$.

Example

Let us look at the alternating series below.

${\sum}_{n = 0}^{\infty} {\left(- 1\right)}^{n} \frac{n}{2 n + 1}$

By Nth Terms (Divergence) Test,

${\lim}_{n \to \infty} \frac{n}{2 n + 1} = {\lim}_{n \to \infty} \frac{1}{2 + \frac{1}{n}} = \frac{1}{2 + 0} = \frac{1}{2} \ne 0$

Hence, the series diverges.

I hope that this was helpful.