### Topics

## Introduction to Series - At A Glance:

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.

#### Example 1

Does the series converge or diverge? | |

We don't have enough information to answer this question yet. We can see that the terms converge to 0, but the Divergence Test only tells us what to do when the terms **don't** converge to 0. | |

#### Exercise 1

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

The series converges.

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

As *n* approaches infinity, n! will overpower (*n* + 1) so

The series might converge, but we don't know for sure. The statement

" converges"

can't be decided using only the Divergence Test.

#### Exercise 2

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

The series diverges.

Answer

For the series, look at the limit of the terms. If

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

As *n* approaches infinity, the exponential 2^{n} overpowers *n*^{2}. This means

so the sequence diverges. The statement

" diverges"

is true.

#### 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.

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

Since 0.9 is less than 1,

This means the statement

" *might* converge"

is true.

#### Exercise 4

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

The series converges.

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

Since 1.1 is greater than 1,

This means we know the series diverges, so the statement

" converges"

is false.

#### Exercise 5

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

The series diverges.

Answer

For the series, look at the limit of the terms. If

,

the series diverges. If

there's hope the series might converge, but we can't tell when our only tool is the Divergence Test.

The limit of the terms is

so the series may or may not converge. Since we're only using the Divergence Test, we can't tell whether the statement

" diverges"

is true or not.

#### Exercise 6

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

Answer

The terms converge to 0, so this series might converge. Further investigation is needed.

#### Exercise 7

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

Answer

This series diverges since the limit of the terms, , diverges.

#### Exercise 8

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

Answer

The limit of the terms is . Since this limit is not equal to 0, the series diverges.

#### Exercise 9

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

Answer

The *n*th term of this series simplifies to . Since , this sequence might converge.

#### Exercise 10

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

Answer

If we write out the *n*th term of this series we get

The *n* and 2 cancel from the numerator and the denominator. This leaves us with an expression that has 2 on the bottom and (*n* – 1) multiplied by a lot of stuff in the top:

As *n* approaches ∞, the terms given by this expression will also approach infinity. Thus the series diverges.