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Chapter 7: Q. 3 (page 655)

Fill in the blanks to complete each of the following theorem statements.

Tests for Monotonicity: A sequence {ak} is increasing if it passes any of the following tests:

The Derivative Test: ____≥0forallx≥1 given that a(x) is a function that is _______on [1,∞) and whose value at any positive integer k isa(k)=____

Short Answer

Expert verified

The required answer is a'(x)≥0for all x≥1given that a(x) is a function that is differentiable on [1,∞)and whose value at any positive integer k isa(k)=ak

Step by step solution

01

Step 1. Given Information 

The given term is Derivative test.

02

Step 2. Explanation 

The derivative test is based on the test for increasing behavior in differentiable functions.

Thus,

a'(x)≥0forallx≥1given that a(x) is a function that is differentiable on [1,∞)and whose value at any positive integer k isa(k)=ak

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Most popular questions from this chapter

Leila, in her capacity as a population biologist in Idaho, is trying to figure out how many salmon a local hatchery should release annually in order to revitalize the fishery. She knows that ifpksalmon spawn in Redfish Lake in a given year, then only 0.2pkfish will return to the lake from the offspring of that run, because of all the dams on the rivers between the sea and the lake. Thus, if she adds the spawn from h fish, from a hatchery, then the number of fish that return from that run k will be pk+1=0.2(pk+h)..

(a) Show that the sustained number of fish returning approaches p∞=h∑k+1∞0.2kas k→∞.

(b) Evaluate p∞.

(c) How should Leila choose h, the number of hatchery fish to raise in order to hold the number of fish returning in each run at some constant P?

In Exercises 48–51 find all values of p so that the series converges.

∑k=2∞1klnkp

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.

Iflimk→∞akbk=∞and∑k=1∞akconverges, explain why we cannot draw any conclusions about the behavior of∑k=1∞bk.

In Exercises 48–51 find all values of p so that the series converges.

∑k=1∞lnkkp
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