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

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Answer 1 · 2014-11-26T18:55:06

If an alternating series has the form

$\infty \sum n = 0 {(- 1)}^{n} b_{n}$ $sum_{n=0}^infty(-1)^nb_n$ ,

where $b_{n} \geq 0$ $b_n ge 0$ ,

then the series diverges if

$lim_{n to infty}b_n ne0$ .

Example

Let us look at the alternating series below.

$sum_{n=0}^infty (-1)^n{n}/{2n+1}$

By Nth Terms (Divergence) Test,

$lim_{n to infty}{n}/{2n+1}=lim_{n to infty}{1}/{2+1/n}=1/{2+0}=1/2 ne 0$

Hence, the series diverges.

I hope that this was helpful.