AP Calculus BC : Ratio Test and Comparing Series

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

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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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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 

Explanation:

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. 

Explanation:

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. 

Explanation:

The series converges. 

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