Nth Term Test for Divergence of Series

We have gone over infinite series and how to expand them, however it can be very tedious to expand every series to determine whether it converges or diverges. There must be a quicker way! Luckily, there are tests we can use, such as the Nth term test for divergence.

##$N$th term test for the divergence of a series

We use the $N$th term test for divergence to determine whether a series diverges or not.

According to the $N$th term test for divergence...

In other words, if the limit of $A_{n}$ does not equal zero, we know the series diverges.

However, we cannot switch the order of this statement!

Be careful!! If the limit of $A_{n}$ does equal zero, we cannot assume the series is convergent. If the limit of $A_{n}$ does equal zero, we have to conduct more tests to determine whether the series diverges or converges, such as the geometric series test.

Let's work through an example problem!

If we are given the series...

We can use the $N$th term test for divergence to determine whether this series diverges or not. Let's take the limit of this series to see.

As the limit equals $\frac{4}{3}$, we know according to the $N$th term test for divergence that the series diverges.

Looking Further

Your next thought might be: what about a test of convergence! You're right! There is one. We can use the ratio test to test for convergence.

The $Nth$ Term Test for Divergence

Math has a simple test to figure out whether a series diverges or converges.

(Video) The nth Term for Test for Divergence

by Kimberly Brehm

In this video by Kimberly Brehm, you will learn how to use the nth term test to check for divergence.