AP Calculus BC : Ratio Test and Comparing Series

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #71 : Ratio Test

Using the Ratio Test, determine what the following series converges to, and whether the series is Divergent, Convergent or Neither.

Possible Answers:

, and Convergent

, and Divergent

, and Neither

, and Neither

, and Divergent

Correct answer:

, and 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 hence 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

We can simplify the expression to

 

When we evaluate the limit, we get.

.

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

Example Question #93 : Polynomial Approximations And Series

Determine what the following series converges to, and whether the series is Convergent, Divergent or Neither.

 

Possible Answers:

, and Convergent

, and Convergent

, and Neither

, and Divergent

, and Neither

Correct answer:

, and Convergent

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 hence 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

We can simplify the expression to

 

When we evaluate the limit, we get.

.

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

Example Question #94 : Polynomial Approximations And Series

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

Possible Answers:

, and Neither

, and Divergent

, and Convergent

, and Convergent

, and Divergent

Correct answer:

, and 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 hence 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

We can simplify the expression to

 

When we evaluate the limit, we get.

.

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

Example Question #95 : Polynomial Approximations And Series

Determine if the following series is Convergent, Divergent or Neither.

Possible Answers:

Divergent

Not enough information.

Convergent

More tests are needed.

Neither

Correct answer:

Convergent

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 hence 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

We can simplify the expression to

 

When we evaluate the limit, we get.

.

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

Example Question #1 : Comparing Series

We know that :
 and

We consider the series having the general term:

Determine the nature of the series:

 

Possible Answers:

The series is divergent.

It will stop converging after a certain number.

The series is convergent.

Correct answer:

The series is convergent.

Explanation:

We know that:

and therefore we deduce :

We will use the Comparison Test with this problem. To do this we will look at the function in general form 

We can do this since,

 and   approach zero as n approaches infinity. The limit of our function becomes,

 

This last part gives us .

Now we know that and noting that is a geometric series that is convergent.

We deduce by the Comparison Test that the series

having general term is convergent.

 

 

Example Question #97 : Polynomial Approximations And Series

We consider the following series:

Determine the nature of the convergence of the series.

Possible Answers:

The series is divergent.

Correct answer:

The series is divergent.

Explanation:

We will use the comparison test to prove this result. We must note the following:

  is positive.

 

We have all natural numbers n:

 , this implies that

.

Inverting we get :

Summing from 1 to , we have

 

We know that the is divergent. Therefore by the comparison test:

 

is divergent

Example Question #11 : Ratio Test And Comparing Series

Using the Limit Test, determine the nature of the series:

Possible Answers:

The series is convergent.

The series is divergent.

Correct answer:

The series is convergent.

Explanation:

We will use the Limit Comparison Test to study the nature of the series.

We note first that , the series is positive.

We will compare the general term to

We note that by letting  and , we have:

.

Therefore the two series have the same nature, (they either converge or diverge at the same time). 

We will use the Integral Test to deduce that the series having the general term:

is convergent.

 

Note that we know that is convergent if p>1 and in our case p=8 .

This shows that the series having general term is convergent.

By the Limit Test, the series having general term  is convergent.

This shows that our series is convergent.

Example Question #99 : Polynomial Approximations And Series

We consider the following series:

Determine the nature of the convergence of the series.

Possible Answers:

The series is divergent.

Correct answer:

The series is divergent.

Explanation:

We will use the Comparison Test to prove this result. We must note the following:

  is positive. 

We have all natural numbers n:

, this implies that

.

Inverting we get :

Summing from 1 to , we have

We know that the is divergent. Therefore by the Comparison Test:

 is divergent.

Example Question #100 : Polynomial Approximations And Series

Is the series

convergent or divergent, and why?

Possible Answers:

Convergent, by the comparison test.

Divergent, by the test for divergence.

Convergent, by the ratio test. 

Divergent, by the comparison test.

Divergent, by the ratio test.

Correct answer:

Convergent, by the comparison test.

Explanation:

We will use the comparison test to prove that

converges (Note: we cannot use the ratio test, because then the ratio will be , which means the test is inconclusive).

We will compare  to  because they "behave" somewhat similarly. Both series are nonzero for all , so one of the conditions is satisfied.

The series 

converges, so we must show that 

 

for .

This is easy to show because

since the denominator  is greater than or equal to  for all .

Thus, since 

and because

converges, it follows that 

converges, by comparison test.

Example Question #1 : Ratio Test

Which of these series cannot be tested for convergence/divergence properly using the ratio test? (Which of these series fails the ratio test?)

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

The ratio test fails when . Otherwise the series converges absolutely if , and diverges if .

Testing the series , we have





Hence the ratio test fails here. (It is likely obvious to the reader that this series diverges already. However, we must remember that all intuition in mathematics requires rigorous justification. We are attempting that here.)

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