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 #1 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Possible Answers:

Convergent

Divergent

Both

Inconclusive

Neither

Correct answer:

Convergent

Explanation:

In order to figure out if 

is divergent, convergent or neither, we need to use the ratio test.

Remember that the ratio test is as follows.

Suppose we have a series . We define,

 

Then if 

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let  

and

Now 

Now lets simplify this expression to 

.

Since 

.

We have sufficient evidence to conclude that the series is convergent.

Example Question #3 : Limits Of Sequences

Determine if the following series is divergent, convergent or neither.

 

Possible Answers:

Neither

Convergent

Divergent

Both

Inconclusive

Correct answer:

Divergent

Explanation:

In order to figure if 

is convergent, divergent or neither, we need to use the ratio test.

Remember that the ratio test is as follows.

Suppose we have a series . We define,

Then if 

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let  

and

Now 

.

Now lets simplify this expression to 

.

Since ,

we have sufficient evidence to conclude that the series is divergent.

 

Example Question #21 : Series Of Constants

Determine if the following series is divergent, convergent or neither.

Possible Answers:

Neither

Divergent

Both

Convergent

Inconclusive

Correct answer:

Divergent

Explanation:

In order to figure if 

is convergent, divergent or neither, we need to use the ratio test. 

Remember that the ratio test is as follows.

Suppose we have a series . We define,

Then if 

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let  

and

.

Now 

.

Now lets simplify this expression to 

.

Since ,

we have sufficient evidence to conclude that the series is divergent.

 

Example Question #22 : Series Of Constants

Determine if the following series is convergent, divergent or neither.

Possible Answers:

Divergent

More tests are needed.

Convergent

Inconclusive

Neither

Correct answer:

Divergent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

  the series is absolutely convergent (and therefore convergent).

  the series is divergent.

 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series diverges.

Example Question #23 : Series Of Constants

Determine if the following series is divergent, convergent or neither.

Possible Answers:

More tests are needed.

Convegent

Divergent

Inconclusive

Neither

Correct answer:

Convegent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

  the series is absolutely convergent (and thus convergent).

  the series is divergent.

 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

.

Now lets simplify this.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series converges.

Example Question #24 : Series Of Constants

Determine if the following series is convergent, divergent or neither.

Possible Answers:

Divergent

Neither

More tests needed.

Convergent

Inconclusive

Correct answer:

Divergent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

  the series is absolutely convergent (therefore convergent).

  the series is divergent.

 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series diverges.

Example Question #6 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Possible Answers:

Inconclusive

Neither

More tests are needed.

Divergent

Convergent

Correct answer:

Divergent

Explanation:

To determine if

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

  the series is absolutely convergent (and thus convergent).

  the series is divergent.

 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can simplify the expression to be

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series diverges.

Example Question #7 : Ratio Test And Comparing Series

Determine of the following series is convergent, divergent or neither.

Possible Answers:

Inconclusive.

Convergent

Divergent

Neither

More tests are needed.

Correct answer:

Divergent

Explanation:

To determine whether this series is convergent, divergent or neither

we need to remember the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

  the series is absolutely convergent (and therefore convergent).

  the series is divergent.

 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this to.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series is divergent.

Example Question #8 : Ratio Test And Comparing Series

Determine what the following series converges to using the ratio test and whether the series is convergent, divergent or neither.

Possible Answers:

, and convergent.

, and convergent.

, and neither.

, and neither.

, and divergent.

Correct answer:

, and convergent.

Explanation:

To determine whether this series is convergent, divergent or neither

we need to remember the ratio test.

The ratio test is as follows.

Suppose we a series  . Then we define,

.

If

  the series is absolutely convergent (thus convergent).

  the series is divergent.

 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

and

Now

We can rearrange the expression to be

Now lets simplify this to.

When we evaluate the limit, we get.

.

Since , we have sufficient evidence to conclude that the series is convergent.

Example Question #2 : Ratio Test And Comparing Series

Determine the convergence or divergence of the following series:

Possible Answers:

The series (absolutely) convergent.

The series may be divergent, conditionally convergent, or absolutely convergent.

The series is divergent.

The series is conditionally convergent.

Correct answer:

The series (absolutely) convergent.

Explanation:

To determine the convergence or divergence of this series, we use the Ratio Test:

If , then the series is absolutely convergent (convergent)

If , then the series is divergent

If , the series may be divergent, conditionally convergent, or absolutely convergent

So, we evaluate the limit according to the formula above:

which simplified becomes

Further simplification results in

Therefore, the series is absolutely convergent.

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