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

Question

Luke Alvarez · Accepted Answer

If an alternating series has the form

#sum_{n=0}^infty(-1)^nb_n#,

where #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.

Samantha Adkison · Answer

undefined To use the Nth term test for divergence on an alternating series, follow these steps:

1. Evaluate the limit as $n$ approaches infinity of the absolute value of the $n$th term of the series.
2. If the limit is not zero, the series diverges.
3. If the limit is zero, the test is inconclusive, and other tests or methods may need to be used to determine convergence or divergence.

Mathematically, the Nth term test for divergence on an alternating series can be stated as follows:

Given an alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$, where $a_n$ is the $n$th term of the series, the series diverges if:

$$\lim_{n \to \infty} |a_n| \neq 0$$

Otherwise, if the limit is zero:

$$\lim_{n \to \infty} |a_n| = 0$$

then the test is inconclusive, and other methods or tests need to be employed to determine convergence or divergence.