/*! 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 Explain why the integral test ma... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

Log out

Learning Materials

Features

Discover

Chapter 7: Q. 31 (page 625)

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

∑k=3∞ 1(k−2)2

Short Answer

Expert verified

Ans: The series ∑k=3∞ 1(k−2)2is convergent and converges to1.

Step by step solution

01

Step 1. Given information.

given,

∑k=3∞ 1(k−2)2

02

Step 2. The objective is to explain why the integral test is used to determine the convergence or divergence of the series and use the test to determine the convergence or divergence of the series.   

Consider function f(x)=1(x−2)2.

The function f(x)=1(x−2)2is continuous, decreasing, with positive terms. Therefore, all the conditions of the integral test are fulfilled. So, the integral test is applicable.

03

Step 3. Consider the integral ∫x=3∞ f(x)dx=∫x=3∞ 1(x−2)2dx.

Therefore,

∫x=3∞ f(x)dx=limk→∞ ∫x=3k 1(x−2)2dx=limk→∞ ∫u=1k−2 1u2du(Putx−2=u⇒dx=du)=limk→∞ −1u1k−2(Integrating)=limk→∞ −1k−2+1(Substitution)=1

04

Step 4. Thus, the value of the integral is ∫x=3∞ 1(x−2)2dx=1

The integral converges. Therefore, the series ∑k=3∞ 1(k−2)2is convergent.

Hence, by integral test, the series ∑k=3∞ 1(k−2)2 is convergent and converges to1.

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 91Ó°ÊÓ!

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

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=1∞akin which ak→0.

(b) A divergent p-series.

(c) A convergent p-series.

Let∑k=0∞crkand∑k=0∞bvk be two convergent geometric series. If b and v are both nonzero, prove that ∑k=0∞crkbvk is a geometric series. What condition(s) must be met for this series to converge?

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.

35.∑k=1∞k2+k2

Consider the series

Fill in the blanks and select the correct word:

Iflimk→∞akbk=∞and∑k=1∞_____divergesthen∑k=1∞_____(converges/diverges).

Let ∑k=1∞akbeaconvergentseriesand∑k=1∞bkbeadivergentseries.Prove that the series ∑k=1∞ak+bkdiverges.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision 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.