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Chapter 7: Q. 8 (page 603)

In Exercises 4鈥�11, give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two divergent sequences {ak}and {bk}such that the sequence {akbk}converges.

Short Answer

Expert verified

The example of the sequence is {ak}={k}and {bk}={k2}.

Step by step solution

01

Step 1. Given Information.

Two divergent sequence {ak}and{bk}.

02

Step 2. Consider the divergent sequence.

Consider the sequence {ak}={k}which is strictly increasing an not bounded above.

So it is divergent sequence.

Consider the sequence {bk}={k2}which is strictly increasing an not bounded above.

So it is divergent sequence.

03

Step 3. Division of the sequence.

The sequence {akbk}={1k}which is a decreasing sequence and bounded below.

And the sequence converges to zero.

So the division of two divergent sequence can be convergent.

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Most popular questions from this chapter

Whenever a certain ball is dropped, it always rebounds to a height60% of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of 1 meter?

Use either the divergence test or the integral test to determine whether the series in Exercises 32鈥�43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

37.鈭�k=1鈭�lnkk

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

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(b) A divergent p-series.

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