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Chapter 7: Q. 13 (page 603)

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

A convergent sequence that is not eventually monotonic.

Short Answer

Expert verified

Examples of the sequences is -1kk+6.

Step by step solution

01

Step 1. Given information.

Consider the given question,

A convergent sequence that is not eventually monotonic.

02

Step 2. Take the sequence -1kk+6.

Consider the sequence -1kk+6.

In the sequence ak=-1kk+6the general term is given below,

ak=-1kk+6

The sequence ak=-1kk+6is not a monotonic sequence because sign of -1kk+6 varies alternately as k increases.

Hence, the given sequence is not a monotonic sequence.

03

Step 3. Take the sequence 

The sequence ak=-1kk+6is a bounded sequence because 0≤ak≤17for k>0.

The given sequence has upper and lower bounds, then the sequence is bounded.

The limit of the sequence is ak=-1kk+6,

localid="1649178846922" limk→∞a=limk→∞-1kk+6=0

The sequenceak=-1kk+6is not monotonic but bounded.

Hence, the sequence converges to0.

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Most popular questions from this chapter

Let αbe any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to α. (Hint: Argue that if you add up some finite number of the terms of ∑k=1∞ 12k−1, the sum will be greater than α. Then argue that, by adding in some other finite number of the terms of

∑k=1∞ −12k , you can get the sum to be less than α. By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to α.)

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

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Iflimk→∞akbk=∞and∑k=1∞_____Convergesthen∑k=1∞_____(converges/diverges).

Given a series ∑k=1∞ak, in general the divergence test is inconclusive when . For a ak→0geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

For each series in Exercises 44–47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder,R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so thatRn≤10-6

∑k=1∞1k2
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