Chapter 11.1, Problem 61E

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Chapter
Section

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

# Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 739 for advice on graphing sequences.)61. a n = n 2 cos n 1 + n 2

To determine

Whether the sequence is convergent or divergent; guess the value of the limit if convergent with valid proof.

Explanation

Given:

The sequence is an=n2cosn1+n2 .

Definition used:

If an is a sequence and limnâ†’âˆžan exists, then the sequence an is said to be convergent; otherwise it is divergent.

Calculation:

Obtain the first 20 terms of the sequence.

 n n21+n2 cosn an=n21+n2â‹…cosn 1 121+12=0.5000 cos1=0.5403 121+12â‹…cos1=0.2702 2 221+22=0.8000 cos2=âˆ’0.4161 221+22â‹…cos2=âˆ’0.3329 3 321+32=0.9000 cos3=âˆ’0.9900 321+32â‹…cos3=âˆ’0.8910 4 421+42=0.9412 cos4=âˆ’0.6536 421+42â‹…cos4=âˆ’0.6152 5 521+52=0.9615 cos5=0.2837 521+52â‹…cos5=0.2728 6 621+62=0.9730 cos6=0.9602 621+62â‹…cos6=0.9343 7 721+72=0.9800 cos7=0.7539 721+72â‹…cos7=0.7388 8 821+82=0.9846 cos8=âˆ’0.1455 821+82â‹…cos8=âˆ’0.1433 9 921+92=0.9878 cos9=âˆ’0.9111 921+92â‹…cos9=âˆ’0.9000 10 1021+102=0.9901 cos10=âˆ’0.8391 1021+102â‹…cos10=âˆ’0.8308 11 1121+112=0.9918 cos11=0.0044 1121+112â‹…cos11=0.0044 12 1221+122=0.9931 cos12=0.8439 1221+122â‹…cos12=0.8381 13 1321+132=0.9941 cos13=0.9074 1321+132â‹…cos13=0.9020 14 1421+142=0.9949 cos14=0.1367 1421+142â‹…cos14=0.1360 15 1521+152=0.9956 cos15=âˆ’0.7597 1521+152â‹…cos15=âˆ’0.7564 16 1621+162=0.9961 cos16=âˆ’0.9577 1621+162â‹…cos16=âˆ’0.9540 17 1721+172=0.9966 cos17=âˆ’0.2752 1721+172â‹…cos17=âˆ’0.2743 18 1821+182=0.9969 cos18=0.6603 1821+182â‹…cos18=0.6583 19 1921+192=0.9972 cos19=0.9887 1921+192â‹…cos19=0.9860 20 2021+202=0.9975 cos20=0.4081 2021+202â‹…cos20=0.4071

Plot the points (n,an),Â forÂ n=1,2,...20 on the graph as shown below in Figure 1.

From the graph, it is observed that the sequence is divergent because the sequence oscillates between âˆ’1 and 1.

To prove:

The sequence an=n2cosn1+n2 is divergent.

Proof:

Obtain the limit of the sequence (the value of the term an as n tends to infinity).

That is, compute limnâ†’âˆžan=limnâ†’âˆžn2cosn1+n2

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