- Topics At a Glance
- Series: This is the Sum That Doesn't End
- Sigma Notation
- Alternating Series
- Convergence of Series
- Finally, Meaning...and Food
- Relationship Between Sequences
- Math-e-magics?
- When Limits of Summation Don't Matter
- Properties of Series
- Special Cases
- Arithmetic Series
- Geometric Series
- Finite Geometric Series
- Infinite Geometric Series
- Decimal Expansion
- Word Problems
- Visualization of Series
- When Limits of Summation Don't Matter
**Tests for Convergence****The Divergence Test**- The Alternating Series Test
- The Ratio Test
- The Integral Test
- The Comparison Test
- Absolute Convergence vs. Conditional Convergence
- Summary of Tests
- Taylor and Maclaurin Series
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.

Does the series

converge or diverge?

Answer.

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

the series diverges.

Example 1

Does the series converge or diverge? |

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.