### All AP Calculus BC Resources

## Example Questions

### Example Question #2 : Ratio Test

Assuming that , . Using the ratio test, what can we say about the series:

**Possible Answers:**

We cannot conclude when we use the ratio test.

It is convergent.

**Correct answer:**

We cannot conclude when we use the ratio test.

As required by this question we will have to use the ratio test. if L<1 the series converges absolutely, L>1 the series diverges, and if L=1 the series could either converge or diverge.

To do so, we will need to compute : . In our case:

Therefore

.

We know that

This means that

Since L=1 by the ratio test, we can't conclude about the convergence of the series.

### Example Question #4 : Ratio Test

Using the ratio test,

what can we say about the series.

where is an integer that satisfies:

**Possible Answers:**

We can't use the ratio test to study this series.

We can't conclude when we use the ratio test.

**Correct answer:**

We can't conclude when we use the ratio test.

Let be the general term of the series. We will use the ratio test to check the convergence of the series.

The Ratio Test states:

then if,

1) L<1 the series converges absolutely.

2) L>1 the series diverges.

3) L=1 the series either converges or diverges.

Therefore we need to evaluate,

we have,

therefore:

.

We know that

and therefore,

This means that :

By the ratio test we can't conclude about the nature of the series. We will have to use another test.

### Example Question #5 : Ratio Test

Consider the following series :

where is given by:

. Using the ratio test, find the nature of the series.

**Possible Answers:**

We can't conclude when using the ratio test.

The series is convergent.

**Correct answer:**

We can't conclude when using the ratio test.

Let be the general term of the series. We will use the ratio test to check the convergence of the series.

if L<1 the series converges absolutely, L>1 the series diverges, and if L=1 the series could either converge or diverge.

We need to evaluate,

we have:

.

Therefore:

. We know that,

and therefore

This means that :

.

By the ratio test we can't conclude about the nature of the series. We will have to use another test.

### Example Question #13 : Convergence And Divergence

We consider the series,

.

Using the ratio test, what can we conclude about the nature of convergence of this series?

**Possible Answers:**

We can't use the ratio test here.

We will need to know the values of to decide.

The series converges to .

The series is convergent.

The series is divergent.

**Correct answer:**

The series is convergent.

Note that the series is positive.

As it is required we will use the ratio test to check for the nature of the series.

We have .

Therefore,

if L>1 the series diverges, if L<1 the series converges absolutely, and if L=1 the series may either converge or diverge.

Since the ratio test concludes that the series converges absolutely.

### Example Question #21 : Ratio Test And Comparing Series

Use the ratio test to determine if the series diverges or converges:

**Possible Answers:**

The series diverges.

Unable to determine.

The series converges.

**Correct answer:**

The series diverges.

This limit is infinite, so the series diverges.

### Example Question #22 : Ratio Test And Comparing Series

Use the ratio test to determine if this series diverges or converges:

**Possible Answers:**

The series diverges

The series converges

Unable to determine

**Correct answer:**

The series converges

Since the limit is less than 1, the series converges.

### Example Question #23 : Ratio Test And Comparing Series

Use the ratio test to determine if the series converges or diverges.

**Possible Answers:**

Unable to determine

The series diverges.

The series converges.

**Correct answer:**

The series diverges.

The series diverges.

### Example Question #24 : Ratio Test And Comparing Series

Use the ratio test to determine if the series diverges or converges:

**Possible Answers:**

The series diverges.

The series converges.

Unable to determine

**Correct answer:**

The series converges.

The series converges.

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