Calculus 2 : Comparing Series

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #2956 : Calculus Ii

Test for convergence

Possible Answers:

Converges absolutely

Cannot be determined

Converges in an interval

Converges conditionally

Diverges

Correct answer:

Converges absolutely

Explanation:

Step 1: Recall the convergence rule of the power series:

According to the convergence rule of the power series....

 converges as long as 

 Step 2: Compare the exponent:

Since , it is greater than . Hence the series converges.

Step 3: Conclusion of the convergence rule

Now, notice that the series isn't an alternating series, so it doesn't matter whether we check for absolute or conditional convergence.

Example Question #2957 : Calculus Ii

Test for convergence

Possible Answers:

Converges absolutely

Converges in an interval

Can't be determined

Converges Conditionally

Diverges

Correct answer:

Converges absolutely

Explanation:

Step 1: Try and look for another function that is similar to the original function:

 looks like 

Step 2: We will now Use the Limit Comparison test

Since the limit calculated, is not equal to 0, the given series converges by limit comparison test

Example Question #2958 : Calculus Ii

Does the following series converge or diverge?

Possible Answers:

Diverge

Conditionally converge

The series either absolutely converges, conditionally converges, or diverges.

Absolutely converge

Correct answer:

Absolutely converge

Explanation:

The best way to answer this question would be by comparing the series to another series,, that greatly resembles the behavior of the original series, . The behavior is determined by the terms of the numerator and the denominator that approach infinity at the quickest rate. In this case:

When this series is simplifies, it simplifies to a series that converges because of the p-test where .

With two series and the confirmed convergence of one of those series, the limit comparison test can be applied to test for the convergence or divergence of the original series. The limit comparison test states that two series will converge or diverge together if:

Specifically:

 

This limit equals one because of the fact that:

  if the coefficients come from the same power. 

Because the limit is larger than zero,  and  will converge or diverge together. Since it was already established that  converges, the original seies, , converges by the limit comparison test.

 

 

Example Question #2959 : Calculus Ii

Determine if the following series converges or diverges:

Possible Answers:

Series diverges

Series converges

Correct answer:

Series converges

Explanation:

 for all ;     is a sum of geometric sequence with base 1/3.

Therefore, said sum converges.

 

Then, by comparison test,    also converges.

Example Question #2960 : Calculus Ii

What can be said about the convergence of the series ?

Possible Answers:

Diverges 

Converges

Inconclusive 

Correct answer:

Converges

Explanation:

Since  for all n>0, and  converges as a P-Series, we may conclude that  must also converge by the comparison test.

Example Question #171 : Series In Calculus

If  and , and it may be said that  converges, what may be said about ?

Possible Answers:

 Converges by the ratio test.

 Diverges by the ratio test.

 Diverges by the comparison test.

 Converges by the comparison test.

 Converges by the test for convergence of geometric series.

Correct answer:

 Converges by the comparison test.

Explanation:

Given two series, 

 and  

where 

 converges,

the Comparison Test states that the second series 

 must also converge, if and only if, it is smaller than the first.

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