Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For each series in Exercises 44–47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder,R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so thatRn10-6

k=11k2

Determine whether the series k=02k+25k-1converges or diverges. Give the sum of the convergent series.

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1,)such that limxf(x)=α>0, What can the integral tells us about the seriesk=1f(k) ?

What is the contrapositive of the implication “If A, then B"?

Find the contrapositives of the following implications:

If a divides b and b dividesc, then a divides c.

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?

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free