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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.

40.k=1sin1kk2

Short Answer

Expert verified

The series is convergent.

Step by step solution

01

Step 1. Given information

We have been given the series k=1sin1kk2

We have to determine whether the series converge or diverge.

02

Step 2. Determine whether the series converge or diverge.

Consider functionfx=sin1xx2

The function is continuous, decreasing on [1,), with positive terms.

All the conditions of integral test are fulfilled.

So, integral test is applicable.

Consider the integral localid="1649093868727" x=1fxdx=x=1sin1xx2dx

localid="1649093922985" x=1fxdx=limkx=1ksin1xx2dx=limkcos1k-cos1=1-cos1

The integral converges.

So, the series is convergent and converges to1-cos1.

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Most popular questions from this chapter

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?

Whenever a certain ball is dropped, it always rebounds to a height p% (0 < p < 100) of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of h meters?

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series k=1akin which ak0.

(b) A divergent p-series.

(c) A convergent p-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=11k

Explain how you could adapt the integral test to analyze a series k=1f(k)in which the functionf:[1,) is continuous, negative, and increasing.

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