Calculus 2 : Alternating Series

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #51 : Types Of Series

Determine if the following series is convergent or divergent: 

Possible Answers:

Convergent according to the alternating series test

Divergent according to the alternating series test

Inconclusive according to the alternating series test

Divergent according to ratio test

Correct answer:

Inconclusive according to the alternating series test

Explanation:

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

Our tests stop here. Since the limit of  goes to infinity, we can say that this function does not converge according to the alternating series test.

Example Question #11 : Alternating Series

Determine if the following series is convergent or divergent using the alternating series test: 

Possible Answers:

Divergent according to the ratio test

Convergent according to the alternating series test

Divergent according to the alternating series test

Inconclusive according to the alternating series test

Correct answer:

Inconclusive according to the alternating series test

Explanation:

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

This concludes our testing. The limit does not equal to 0, so we cannot say that this series converges according to the alternate series test.

Example Question #53 : Types Of Series

Determine if the following series is convergent or divergent using the alternating series test: 

Possible Answers:

Divergent according to the ratio test

Convergent according to the alternate series test

Divergent according to the alternate series test

Inconclusive according to the alternate series test

Correct answer:

Convergent according to the alternate series test

Explanation:

Conclusive according to the alternating series test

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

Example Question #54 : Types Of Series

Determine if the following series is convergent or divergent using the alternating series test: 

Possible Answers:

Inconclusive according to the alternating series test

Divergent according to the ratio test

Divergent according to the alternating series test

Convergent according to the alternating series test

Correct answer:

Convergent according to the alternating series test

Explanation:

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

Example Question #55 : Types Of Series

Determine if the following series is convergent or divergent using the alternating series test: 

Possible Answers:

Inconclusive according to the alternating series test

Divergent according to the alternating series test

Convergent according to the alternating series test

Divergent according to the ratio test

Correct answer:

Inconclusive according to the alternating series test

Explanation:

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

 

1.

Conclude with tests. The limit of  goes to infinity and so we cannot use the alternating series test to reach a convergence/divergence conclusion.

Example Question #16 : Alternating Series

Determine if the following series is convergent or divergent using the alternating series test: 

Possible Answers:

Convergent according to the alternating series test

Divergent according to the alternating series test

Inconclusive according to the alternating series test

Divergent according to the ratio test

Correct answer:

Convergent according to the alternating series test

Explanation:

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

Example Question #17 : Alternating Series

Determine if the following series is convergent or divergent using the alternating series test: 

Possible Answers:

Convergent according to the alternating series test

Divergent according to the alternating series test

Inconclusive according to the alternating series test

Divergent according to the ratio test

Correct answer:

Convergent according to the alternating series test

Explanation:

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

Example Question #18 : Alternating Series

Determine how many terms need to be added to approximate the following series within

Possible Answers:

Correct answer:

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that 

Step 1:

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

2. {} is a decreasing sequence

Since the factorial is always increasing, the sequence is always decreasing.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until 

so 7 terms are needed to approximate the sum within .001.

 

Example Question #5 : Sequences & Series

Determine how many terms need to be added to approximate the following series within

Possible Answers:

Correct answer:

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that 

Step 1:

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2. {} is a decreasing functon, since a factorial never decreases.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until 

4 needs to be added to approximate the sum within .

Example Question #19 : Alternating Series

Determine how many terms need to be added to approximate the following series within

Possible Answers:

Correct answer:

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that 

Step 1:

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

  1.        
  2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until 

7 terms are needed to approximate the sum within .001

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