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

Find the values of x for which the series K=0sinxkconverges.

Short Answer

Expert verified

The series K=0sinxkconverges for all real numbers except for x2n+1π2|n.

Step by step solution

01

Step 1. Given information.

Given a series K=0sinxk.

02

Step 2. Find all values of x for which the series converges.

A geometric series is of the form k=0crkfor some constants c and r.

Supposer is a non-zero real number, then k=0crk converges to c1-rif and only if r<1.

Here, the series localid="1648832395698" K=0sinxkhas r=sinx.

For the series to converge, sinx<1.

Note that -1sinx1.

So, sinx=1if and only if localid="1648832424318" x=2n+1π2 for all natural number n.

It follows that K=0sinxkconverges for all real number except for x=2n+1π2where localid="1648832538629" n.

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

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=hk+10.2kas 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?

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.

37.k=1lnkk

Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.

1.272727...

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=1k1+kk2

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

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