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Chapter 7: Q. 12 (page 603)

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

Explain what is important about monotonic and bounded sequences.

Short Answer

Expert verified

If the sequence is decreasing and is bounded below then the sequence is convergent and if the decreasing sequence is not bounded below then the sequence is divergent.

If the sequence is increasing and is bounded above then the sequence is convergent and if the increasing sequence is not bounded below then the sequence is divergent.

Step by step solution

01

Step 1. Given information.

Consider the given question,

The sequence is monotonic and bounded.

02

Step 2. Explain the monotonicity and boundedness of the sequence.

Consider the monotonicity and boundedness of the sequence.

The monotonicity and boundedness of the sequence helps in determining the convergence and divergence of the sequence.

If the sequence is decreasing and is bounded below then the sequence is convergent and if the decreasing sequence is not bounded below then the sequence is divergent.

If the sequence is increasing and is bounded above then the sequence is convergent and if the increasing sequence is not bounded below then the sequence is divergent.

Thus, both the monotonicity and boundedness of the sequence is important for determining the behavior of the sequence.

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Most popular questions from this chapter

If limk→∞akbkWhereLis a positive finite number, what may we conclude about the two series?

Prove Theorem 7.24 (a). That is, show that if c is a real number and∑k=1∞ak is a convergent series, then ∑k=1∞cak=c∑k=1∞ak.

Let∑k=0∞crkand∑k=0∞bvk be two convergent geometric series. If b and v are both nonzero, prove that ∑k=0∞crkbvk is a geometric series. What condition(s) must be met for this series to converge?

Determine whether the series ∑k=0∞-3k+14k-2converges or diverges. Give the sum of the convergent 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.
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