Calculus 2 : Concepts of Convergence and Divergence

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #11 : Concepts Of Convergence And Divergence

There are 2 series,  and , and they are both divergent. Is  convergent, divergent, or inconclusive?

Possible Answers:

Divergent

Convergent

Inconclusive

Correct answer:

Inconclusive

Explanation:

 is divergent

 is divergent

However, unlike convergent series in which the sum of convergent series will produce a convergent series, this is not the case for divergent series. Due to the nature of infinite series, adding together 2 divergent series may be divergent, but it may also produce a convergent series. More information is needed.

 is inconclusive

Example Question #12 : Concepts Of Convergence And Divergence

Use the limit test (divergence test) to find if the series is convergent, divergent, or inconclusive.

Possible Answers:

Inconclusive

Convergent

Divergent

Correct answer:

Inconclusive

Explanation:

Divergence Test and Limit Test are the same tests with different names.

If then  diverges.

However, if  the test is inconclusive.

Solution:

The series is inconclusive by the divergence test.

Example Question #13 : Concepts Of Convergence And Divergence

Use the limit test (divergence test) to find if the series is convergent, divergent, or inconclusive.

Possible Answers:

Convergent

Divergent

Inconclusive

Correct answer:

Inconclusive

Explanation:

Divergence Test and Limit Test are the same tests with different names.

If  then  diverges.

However, if  the test is inconclusive.

Solution:

This series is inconclusive by the divergence test.

Example Question #31 : Introduction To Series In Calculus

Use the limit test (divergence test) to find if the series is convergent, divergent, or inconclusive.

Possible Answers:

Convergent

Inconclusive

Divergent

Correct answer:

Divergent

Explanation:

Divergence Test and Limit Test are the same tests with different names.

If  then  diverges.

However, if  the test is inconclusive.

Solution:

The series is divergent by the divergence test.

Example Question #32 : Introduction To Series In Calculus

Use the limit test (divergence test) to find if the series is convergent, divergent, or inconclusive.

Possible Answers:

Convergent

Divergent

Inconclusive

Correct answer:

Inconclusive

Explanation:

Divergence Test and Limit Test are the same tests with different names.

If  then  diverges.

However, if  the test is inconclusive.

Solution:

Although ln(n) also tends to infinity, n grows to infinity more quickly than ln(n), and so the limit will go to 0.

The series is inconclusive by the divergence test.

Example Question #33 : Introduction To Series In Calculus

Does the following series converge absolutely, conditionally, or is it divergent? 

Possible Answers:

Divergent

Conditionally Convergent

Absolutely Convergent

Correct answer:

Absolutely Convergent

Explanation:

A series is absolutely convergent if  is convergent.

If  is not convergent, but  is convergent, then it is conditionally convergent.

This nuance matters when testing series with negatives. First test for absolute convergence.

 is a geometric series, with .

So this series is absolutely convergent by the geometric test.

Example Question #34 : Introduction To Series In Calculus

Does the following series converge absolutely, conditionally, or is it divergent? 

Possible Answers:

Divergent

Conditionally Convergent

Absolutely Convergent

Correct answer:

Divergent

Explanation:

A series is absolutely convergent if  is convergent.

If  is not convergent, but  is convergent, then it is conditionally convergent.

This nuance matters when testing series with negatives. First test for absolute convergence.

This is a geometric series with r > 1, and |r| > 1, so this series is divergent by the geometric test.

Example Question #11 : Concepts Of Convergence And Divergence

Does the following series converge absolutely, conditionally, or is it divergent? 

Possible Answers:

Absolutely Convergent

Conditionally Convergent

Divergent

Correct answer:

Divergent

Explanation:

A series is absolutely convergent if  is convergent.

If  is not convergent, but  is convergent, then it is conditionally convergent.

This nuance matters when testing series with negatives. First test for absolute convergence.

This series diverges by the divergence test.

Now test for conditional convergence.

This limit bounces between positive and negative infinity, and so the limit does not exist. By the divergence test, this series diverges.

 

Example Question #31 : Series In Calculus

Does the following series converge absolutely, conditionally, or is it divergent? 

Possible Answers:

Divergent

Absolutely Convergent

Conditionally Convergent

Correct answer:

Divergent

Explanation:

A series is absolutely convergent if  is convergent.

If is not convergent, but  is convergent, then it is conditionally convergent.

This nuance matters when testing series with negatives. First test for absolute convergence.

The absolute series diverges by the divergence test.

Now test for conditional convergence.

 alternates between positive and negative infinity, and so it diverges by the divergence test.

This series diverges.

Example Question #1 : Limits Of Sequences

There are 2 series and .

Is the sum of these 2 infinite series convergent, divergent, or inconclusive?

Possible Answers:

Inconclusive

Convergent

Divergent

Correct answer:

Convergent

Explanation:

A way to find out if the sum of the 2 infinite series is convergent or not is to find out whether the individual infinite series are convergent or not.

Test the first series 

.

This is a geometric series with .

By the geometric test, this series is convergent.

 

Test the second series 

.

This is a geometric series with .

By the geometric test, this series is convergent.

 

Since both of the series are convergent,  is also convergent. 

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