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Chapter 7: Q. 4 (page 655)

Dominance Relationships for Sequences: Order the following sequences by dominance when a>0andb>1

lnk

Short Answer

Expert verified

The required order of dominance sequence islnk<<ka<<bk<<k!

Step by step solution

01

Step 1. Given information 

The given expression islnk

02

Step 2. Explanation 

Use the concept of dominance relationships for simple sequences,

For any real number a>0andb>1, the following string of dominance holds.

lnk<<ka<<bk<<k!for k sufficiently large.

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Most popular questions from this chapter

Determine whether the series ∑j=2∞2j3converges or diverges. Give the sum of the convergent series.

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.

∑k=1∞ k2+1k!

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.

∑k=1∞ 1kk

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.

What is meant by a p-series?
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