/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q. 31 Use the ratio test for absolute ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q. 31 (page 653)

Use the ratio test for absolute convergence to determine whether the series in Exercises 30鈥�35 converge absolutely or diverge.
鈭�k=0鈭�(-3)k+1(2k)!

Short Answer

Expert verified

The series 鈭�k=0鈭�(-3)k+1(2k)!converges absolutely.

Step by step solution

01

Step 1. Given Information.

The series:

鈭�k=0鈭�(-3)k+1(2k)!

02

Step 2. Rewrite the series.

ak=(-3)k+1(2k)!

03

Step 3. Find ak+1.

ak+1=(-3)k+1+1(2(k+1))!=(-3)k+2(2k+2)!

04

Step 4. Calculate ak+1ak.

ak+1ak=(-3)k+2(2k+2)!(-3)k+1(2k)!=(-3)k+2(2k)!(-3)k+1(2k+2)!=-3(-3)k+1(2k)!(-3)k+1(2k+2)(2k+1)(2k)!=-3(2k+2)(2k+1)=32(k+1)(2k+1)

05

Step 5. Take limits.

limk鈫�鈭�ak+1ak=limk鈫�鈭�(32(k+1)(2k+1))=0

So by the ratio test, the series converges absolutely.

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 91影视!

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

Determine whether the series 鈭�k=0鈭�-3k+14k-2converges or diverges. Give the sum of the convergent series.

Given thata0=-3,a1=5,a2=-4,a3=2and鈭�akk=2鈭�=7, find the value ofrole="math" localid="1648828803227" 鈭�akk=3鈭�.

Determine whether the series 鈭�j=2鈭�2j3converges or diverges. Give the sum of the convergent series.

Iflimk鈫�鈭�akbk=鈭�and鈭�k=1鈭�_____Convergesthen鈭�k=1鈭�_____(converges/diverges).

What is the contrapositive of the implication 鈥淚f A, then B"?

Find the contrapositives of the following implications:

If a divides b and b dividesc, then a divides c.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation
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.