/*! 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. 25 Explain why the integral test ma... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

Log out

Learning Materials

Features

Discover

Chapter 7: Q. 25 (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=0∞ e−k∑k=0∞ e−k

Short Answer

Expert verified

Ans: The series∑k=0∞ e−kis convergent.

Step by step solution

01

Step 1. Given information.

given,

∑k=0∞ e−k

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 functionf(x)=e-x.

The first derivative of the functionrole="math" localid="1649067766819" f(x)=e-xis:

f'(x)=-e-x

The derivative is negative for all x>1. Therefore the function is decreasing.

The functionf(x)=e-xis 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=1∞ f(x)dx=∫x=1∞ e−xdx.

Therefore,

∫x=0∞ e−xdx=limk→∞ ∫x=0k e−xdx=limk→∞ −e−xx=0k(Integrate)=limk→∞ −e−k+e0=limk→∞ −e−k+1(Substitution)=0+1=1(Take limit)

04

Step 4. Thus, the value of the integral is ∫x=0∞ e−xdx=1

The integral converges. Therefore, the series ∑k=0∞ e−kis convergent.

Hence, by integral test, the series ∑k=0∞ e−k is convergent.

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

Determine whether the series ∑k=0∞-3k+14k-2converges or diverges. Give the sum of the convergent 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=1∞ k!kk

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=1∞1k2

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.

3.454545...

Determine whether the series ∑n=1∞-1n+13nconverges or diverges. Give the sum of the convergent series.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.