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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: ____0forallx1 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 x1given 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)0forallx1given 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

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=11k

Leila finds that there are more factors affecting the number of salmon that return to Redfish Lake than the dams: There are good years and bad years. These happen at random, but they are more or less cyclical, so she models the number of fish qkreturning each year as qk+1=(0.14(1)k+0.36)(qk+h), where h is the number of fish whose spawn she releases from the hatchery annually.

(a) Show that the sustained number of fish returning in even-numbered years approach approximately qe=3hk=10.11k.

(Hint: Make a new recurrence by using two steps of the one given.)

(b) Show that the sustained number of fish returning in odd-numbered years approaches approximately qo=6111hk=10.11k.

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

Consider the series

Fill in the blanks and select the correct word:

Iflimkakbk=andk=1_____divergesthenk=1_____(converges/diverges).

Provide a more general statement of the integral test in which the function f is continuous and eventually positive, and decreasing. Explain why your statement is valid.

Let 0 < p < 1. Evaluate the limitlimk1/klnk1/kp

Explain why we cannot use a p-series with 0 < p < 1 in a limit comparison test to verify the divergence of the seriesk=21klogk

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