/*! 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. 33 Conditional and absolute converg... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

Log out

Learning Materials

Features

Discover

Chapter 7: Q. 33 (page 657)

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.

∑k=1∞(-1)ke-k

Short Answer

Expert verified

The series ∑k=1∞(-1)ke-k converges absolutely.

Step by step solution

01

Step 1. Given Information.

The series:

∑k=1∞(-1)ke-k

02

Step 2. By Alternating Series Test.

According to the Alternating Series Test, the sequence ak+1<akfor every . Then the alternating series ak+1,akboth converges.

03

Step 3. Find ak+1.

ak=e-kak+1=e-(k+1)=e-k-1ak+1<ak

So the sequence is monotonic decreasing sequence.

04

Step 4. Find limk→∞ak.

limk→∞ak=limk→∞e-k=0

So the series converges.

05

Step 5. Find ak+1ak.

limk→∞ak+1ak=limk→∞(-1)k+1e-k-1(-1)ke-k=limk→∞1eL=1e<1

So 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 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

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.

0.305130513051...

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

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak→0, then ∑k=1∞akconverges.

(b) True or False: If ∑k=1∞akconverges, then ak→0.

(c) True or False: The improper integral ∫1∞f(x)dxconverges if and only if the series ∑k=1∞f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series ∑k=1∞k-pconverges.

(f) True or False: If f(x)→0as x→∞, then ∑k=1∞f(k) converges.

(g) True or False: If ∑k=1∞f(k)converges, then f(x)→0as x→∞.

(h) True or False: If ∑k=1∞ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

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.

0.6345345...

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 thatRn≤10-6

∑k=1∞1k2
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.