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Chapter 7: Q. 10 (page 610)

Briefly outline the advantages and disadvantages of using the two comparison tests to analyze the behavior of a series ∑akk=1∞.

Short Answer

Expert verified

The properties of both tests is given below:

Step by step solution

01

Step 1. Advantages and Disadvantages Comparison Test

Advantages: If you can find the right series to use for comparison, this test always gives an answer.

Disadvantages: If the series does not easily compare to one of our “known” series, then you are out of luck. You also kind of have an intuition about whether or not the series you are testing is convergent or not so you can find the right series for comparison

02

Step 2. Advantages and Disadvantages Limit Comparison Test

Advantages: This test is easier to apply than the comparison test.

Disadvantages: It can also yield inconclusive results, when the limit doesn’t exist or when the cases in the statement of the test are not satisfied.

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Most popular questions from this chapter

Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.

0.99999...

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series ∑k=1∞akin which ak→0.

(b) A divergent p-series.

(c) A convergent p-series.

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak→0, then ∑k=1∞akconverges.

(b) True or False: If ∑k=1∞akconverges, then ak→0.

(c) True or False: The improper integral ∫1∞f(x)dxconverges if and only if the series ∑k=1∞f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series ∑k=1∞k-pconverges.

(f) True or False: If f(x)→0as x→∞, then ∑k=1∞f(k) converges.

(g) True or False: If ∑k=1∞f(k)converges, then f(x)→0as x→∞.

(h) True or False: If ∑k=1∞ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

The contrapositive: What is the contrapositive of the implication “If A, then B.”?

Find the contrapositives of the following implications:

If a quadrilateral is a square, then it is a rectangle.

Consider the series

Fill in the blanks and select the correct word:

Iflimk→∞akbk=∞and∑k=1∞_____divergesthen∑k=1∞_____(converges/diverges).
See all solutions

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