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 #1 : Concepts Of Convergence And Divergence

One of the following infinite series CONVERGES. Which is it?

Possible Answers:

None of the others converge.

Correct answer:

Explanation:

 converges due to the comparison test.

 

We start with the equation . Since  for all values of k, we can multiply both side of the equation by the inequality and get  for all values of k. Since  is a convergent p-series with   hence also converges by the comparison test.

Example Question #2 : Concepts Of Convergence And Divergence

Determine the nature of the following series having the general term:

Possible Answers:

The series is convergent.

Correct answer:

The series is convergent.

Explanation:

We will use the Limit Comparison Test to show this result.

We first denote the genera term of the series by:

 and .

 

We have and the series have the same nature .

We know that

   is convergent by comparing the integral

 

which we know is convergent.

Therefore by the Limit Comparison Test.

we have .

 

Example Question #3 : Concepts Of Convergence And Divergence

If          converges, which of the following statements must be true?

Possible Answers:

For some large value of .

None of the other answers must be true.

The limit of the  term as  approaches infinity is not zero.

The limit of the  partial sums as  approaches infinity is zero.

Correct answer:

For some large value of .

Explanation:

If the series converges, then we know the terms must approach zero. At some point, the terms will be less than 1, meaning when you take the third power of the term, it will be less than the original term.

Other answers are not true for a convergent series by the  term test for divergence.

In addition, the limit of the  partial sums refers to the value the series converges to. A convergent series need not converge to zero. The alternating harmonic series is a good counter example to this.

Example Question #1 : Radius And Interval Of Convergence Of Power Series

Which of following intervals of convergence cannot exist?

Possible Answers:

For any  such that , the interval 

For any , the interval  for some 

Correct answer:

Explanation:

 cannot be an interval of convergence because a theorem states that a radius has to be either nonzero and finite, or infinite (which would imply that it has interval of convergence ). Thus,  can never be an interval of convergence.

Example Question #4 : Concepts Of Convergence And Divergence

Which of the following statements is true regarding the following infinite series?

Possible Answers:

The series diverges, by the divergence test, because the limit of the sequence  does not approach a value as 

The series diverges to .

The series converges because 

The series diverges because  for some  and finite.

Correct answer:

The series diverges, by the divergence test, because the limit of the sequence  does not approach a value as 

Explanation:

The divergence tests states for a series , if  is either nonzero or does not exist, then the series diverges.

The limit  does not exist, so therefore the series diverges. 

Example Question #5 : Concepts Of Convergence And Divergence

Determine whether the following series converges or diverges:

Possible Answers:

The series diverges.

The series conditionally converges.

None of the other answers.

The series converges.

Correct answer:

The series converges.

Explanation:

To prove the series converges, the following must be true:

If converges, then converges.

Now, we simply evaluate the limit:

The shortcut that was used to evaluate the limit as n approaches infinity was that the coefficients of the highest powered term in numerator and denominator were divided.

The limit approaches a number (converges), so the series converges.  

Example Question #1 : Geometric Series

Determine whether the following series converges or diverges. If it converges, what does it converge to? 

Possible Answers:

Correct answer:

Explanation:

First, we reduce the series into a simpler form.

We know this series converges because

By the Geometric Series Theorem, the sum of this series is given by

Example Question #6 : Concepts Of Convergence And Divergence

Determine whether the following series converges or diverges. If it converges, what does it converge to?

Possible Answers:

 

Correct answer:

Explanation:

Notice how this series can be rewritten as

Therefore this series diverges. 

Example Question #1 : Sequences & Series

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

Possible Answers:

Divergent

Inconclusive

Convergent

Correct answer:

Convergent

Explanation:

Infinite series can be added and subtracted with each other.

Since the 2 series are convergent, the sum of the convergent infinite series is also convergent.

 

Note: The starting value, in this case n=1, must be the same before adding infinite series together.

Example Question #2 : Sequences & Series

You have a divergent series  , and you multiply it by a constant 10. Is the new series  convergent or divergent?

Possible Answers:

Convergent

Divergent

Inconclusive

Correct answer:

Divergent

Explanation:

This is a fundamental property of series.

For any constant c, if  is convergent then  is convergent, and if  is divergent,  is divergent.

 

 is divergent in the question, and the constant c is 10 in this case, so  is also divergent.

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