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Chapter 7: Q. 15 (page 603)

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.

A decreasing sequence that is bounded below but is not bounded above.

Short Answer

Expert verified

Examples of the sequences is ak.

Step by step solution

01

Step 1. Given information.

Consider the given question,

A decreasing sequence that is bounded below but is not bounded above.

02

Step 2. Take the sequence ak.

Consider the sequence ak.

The sequence is decreasing if ak+1鈮�akfor k>0.

The decreasing sequence is always bounded above because first term of the sequence is the upper bound of the sequence is decreasing sequence.

So, there is no decreasing sequence which not bounded above.

Hence, this sequence is decreasing sequence that is bounded below but is not bounded above.

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Most popular questions from this chapter

Iflimk鈫�鈭�akbk=鈭�and鈭�k=1鈭�_____Convergesthen鈭�k=1鈭�_____(converges/diverges).

For a convergent series satisfying the conditions of the integral test, why is every remainder Rnpositive? How can Rnbe used along with the term Sn from the sequence of partial sums to understand the quality of the approximation Sn?

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

鈭�k=2鈭�鈥�1kln鈦�k

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

鈭�k=0鈭�鈥�e鈭�k鈭�k=0鈭�鈥�e鈭�k

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

鈭�k=1鈭�鈥�k2+1k!
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