# 17calculus - nth term test

1. n-th Term Test

The nth-Term Test seems to be pretty straight-forward. However, there is a trap that many students fall into.
Basically, it says that, for a series $$\displaystyle{ \sum_{n=1}^{\infty}{a_n}}$$, if $$\displaystyle{ \lim_{n \to \infty}{a_n} \neq 0 },$$ then the series diverges. Pretty simple, eh?

Trap: There is a tendency by almost every student to use this theorem to prove convergence. The statement says nothing about convergence. Let us give you a couple of examples that demonstrate what we mean. Let's compare the convergence or divergence of these two very similar series. (If you don't know about p-series yet, just take our word for the convergence/divergence conclusions. You will understand this soon enough.)

 A. $$\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n}}}$$ $$\displaystyle{\lim_{n\to\infty}{\frac{1}{n}} = 0}$$ diverges p-series with $$p=1$$ B. $$\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^2}}}$$ $$\displaystyle{\lim_{n\to\infty}{\frac{1}{n^2}} = 0}$$ converges p-series with $$p=2$$

It is important to notice that in both cases the limit of the terms goes to zero. However, series A diverges while series B converges.

So you can see that just because the limit goes to zero, this does not guarantee the series will converge. The only time you can apply this theorem is when the limit does not go to zero. This guarantees divergence. When the limit does go to zero, you still don't know if the series converges or diverges. You need to use another test to determine convergence.

Below is the nth-term test row from the infinite series table. Notice that the convergence box is empty. Just to emphasize, this means this test cannot be used for convergence. Think about this thoroughly and completely so that you get your head around it.

Test

Series

Convergence
Condition(s)

Divergence
Condition(s)

Notes

nth-Term

$\sum_{n=1}^{\infty}{a_n }$ $\lim_{n \rightarrow \infty}{ a_n \ne 0 }$

This test can be used only for divergence.

Here is a quick video that explains this test.

# 17calculus - nth term test

2. Practice

Select a practice problem from the list of buttons below.

Instructions - - For each of the following series, determine whether the series converges or diverges.

Basic Level A open-close

Intermediate Level B open-close