Single Variable Calculus: Concepts and Contexts, Enhanced Edition

4th Edition

ISBN: 9781337687805

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 8.2, Problem 5E

To determine

The first 10 partial sums, the graph of the sequence of the terms and the sequence of the partial sums and determine if the series is convergent or divergent and hence find the sum of the series.

Expert Solution & Answer

Answer to Problem 5E

The first 10 partial sums are: $0.44721$ , $1.15432$ , $1.98637$ , $2.88080$ , $3.80927$ , $4.75796$ , $5.71948$ , $6.68962$ , $7.66581$ and $8.64639$ .

The given series is divergent.

Explanation of Solution

Given: The given series is $∑n=1∞nn2+4$ .

Concepts Used: Let $∑n=1∞an$ be any series. Then the $nth$ partial sum of the series is given by-

$sn=∑k=1nak=a1+a2+....+an$

If the sequence ${sn}$ is convergent and $limn→∞sn=s$ as a real number, then the series $∑n=1∞an$ is convergent and the sum of the series is equal to s , i.e.-

$∑n=1∞an=s$

If the sequence ${sn}$ is divergent, then the series is also divergent.

Also, if $limn→∞an$ doesn’t exist or if $limn→∞an≠0$ , then also the series $∑n=1∞an$ is divergent.

Calculations: The first 10 terms of the given sequence are as follows-

$a1=0.447214$

$a2=0.707107$

$a3=0.832050$

$a4=0.894427$

$a5=0.928477$

$a6=0.948683$

$a7=0.961524$

$a8=0.970143$

$a9=0.976187$

$a10=0.980581$

The first 10 partial sums are as follows-

$s1=∑n=11nn2+4=0.44721$

$s2=∑n=12nn2+4=1.15432$

$s3=∑n=13nn2+4=1.98637$

$s4=∑n=14nn2+4=2.88080$

$s5=∑n=15nn2+4=3.80927$

$s6=∑n=16nn2+4=4.75796$

$s7=∑n=17nn2+4=5.71948$

$s8=∑n=18nn2+4=6.68962$

$s9=∑n=19nn2+4=7.66581$

$s10=∑n=110nn2+4=8.64639$

Hence, the graph of the terms of the sequence of the partial sums of the sequence is given by-

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 8.2, Problem 5E

Now,

$limn→∞an=limn→∞nn2+4$

$=limn→∞1n2+4n$

$=limn→∞1n2+4n2$

$=limn→∞11+4n2$

$=1$ (Since $limn→∞4n2=0$ )

Since, $limn→∞an≠0$ , thus, the given series is divergent.

Chapter 8.2, Problem 4E

Chapter 8.2, Problem 6E

Chapter 8 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - Prob. 37E Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - Prob. 43E Ch. 8.1 - Prob. 44E Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.1 - Prob. 47E Ch. 8.1 - Prob. 48E Ch. 8.1 - Prob. 49E Ch. 8.1 - Prob. 50E Ch. 8.1 - Prob. 51E Ch. 8.1 - Prob. 52E Ch. 8.1 - Prob. 53E Ch. 8.1 - Prob. 54E Ch. 8.1 - Prob. 55E Ch. 8.1 - Prob. 56E Ch. 8.1 - Prob. 57E Ch. 8.1 - Prob. 58E Ch. 8.1 - Prob. 59E Ch. 8.1 - Prob. 60E Ch. 8.2 - 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