# At a Glance - The Divergence Test

Since an uncontrolled grilled cheese spill is a hazardous materials catastrophe we'd like to avoid, we want a test that will tell us when *not* to open the Pandora's box.

We know that if a series converges, its terms must approach zero. Rephrasing this in the contrapositive, *if the terms of a series don't approach zero, the series diverges.* This statement lets us look at a series and, if the terms don't approach zero, conclude that the series diverges.

**Be Careful:** We can't use this statement to conclude that a series converges. We can only use it to evaluate if a series diverges. That's why we call it the **Divergence Test**. If the terms do approach zero, there's hope that the series might converge, but we would need to use other tools to really draw that conclusion.

### Sample Problem

Does the series

converge or diverge?

Answer.

Look at the limit of the terms *a _{n}* =

*n*as

*n*goes to ∞. Since

the series diverges.

#### Exercise 1

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series converges.

#### Exercise 2

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series diverges.

#### Exercise 3

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series might converge.

#### Exercise 4

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series converges.

#### Exercise 5

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series diverges.

#### Exercise 6

Use the divergence test to determine if the following series MUST diverge.

#### Exercise 7

Use the divergence test to determine if the following series MUST diverge.

#### Exercise 8

Use the divergence test to determine if the following series MUST diverge.

#### Exercise 9

Use the divergence test to determine if the following series MUST diverge.

#### Exercise 10

Use the divergence test to determine if the following series MUST diverge.