SOLUTION MANUAL
All Chapters Included
, CONTENTS
Chapter 10 INFINITE SERIES (LT Chapter 11) 1 Chapter 14 DIFFERENTIATION IN SEVERAL
VARIABLES (LT Chapter 15) 336
10.1 Sequences (LT Section 11.1) 1
10.2 Summing an Infinite Series (LT Section 11.2) 13 14.1 Functions of Two or More Variables
10.3 Convergence of Series with Positive Terms (LT Section 15.1) 336
(LT Section 11.3) 23 14.2 Limits and Continuity in Several Variables
10.4 Absolute and Conditional Convergence (LT Section 15.2) 345
(LT Section 11.4) 38 14.3 Partial Derivatives (LT Section 15.3) 351
10.5 The Ratio and Root Tests (LT Section 11.5) 44 14.4 Differentiability and Tangent Planes
10.6 Power Series (LT Section 11.6) 52 (LT Section 15.4) 363
10.7 Taylor Series (LT Section 11.7) 64 14.5 The Gradient and Directional Derivatives
Chapter Review Exercises 81 (LT Section 15.5) 374
14.6 The Chain Rule (LT Section 15.6) 386
14.7 Optimization in Several Variables
Chapter 11 PARAMETRIC EQUATIONS, (LT Section 15.7) 399
POLAR COORDINATES, 14.8 Lagrange Multipliers: Optimizing with a Constraint
AND CONIC SECTIONS (LT Section 15.8) 419
(LT Chapter 12) 96 Chapter Review Exercises 442
11.1 Parametric Equations (LT Section 12.1) 96
Chapter 15 MULTIPLE INTEGRATION
11.2 Arc Length and Speed (LT Section 12.2) 112
11.3 Polar Coordinates (LT Section 12.3) 120
(LT Chapter 16) 458
11.4 Area and Arc Length in Polar Coordinates
(LT Section 12.4) 133
15.1 Integration in Two Variables (LT Section 16.1) 458
15.2 Double Integrals over More General Regions
11.5 Conic Sections (LT Section 12.5) 143
(LT Section 16.2) 471
Chapter Review Exercises 154
15.3 Triple Integrals (LT Section 16.3) 489
15.4 Integration in Polar, Cylindrical, and Spherical
Chapter 12 VECTOR GEOMETRY Coordinates (LT Section 16.4) 502
(LT Chapter 13) 166 15.5 Applications of Multiple Integrals
(LT Section 16.5) 521
12.1 Vectors in the Plane (LT Section 13.1) 166 15.6 Change of Variables (LT Section 16.6) 537
12.2 Vectors in Three Dimensions (LT Section 13.2) 176 Chapter Review Exercises 554
12.3 Dot Product and the Angle between Two Vectors
(LT Section 13.3) 184 Chapter 16 LINE AND SURFACE INTEGRALS
12.4 The Cross Product (LT Section 13.4) 198 (LT Chapter 17) 576
12.5 Planes in Three-Space (LT Section 13.5) 212
12.6 A Survey of Quadric Surfaces (LT Section 13.6) 223 16.1 Vector Fields (LT Section 17.1) 576
12.7 Cylindrical and Spherical Coordinates 16.2 Line Integrals (LT Section 17.2) 581
(LT Section 13.7) 230 16.3 Conservative Vector Fields (LT Section 17.3) 599
Chapter Review Exercises 240 16.4 Parametrized Surfaces and Surface Integrals
(LT Section 17.4) 606
16.5 Surface Integrals of Vector Fields
Chapter 13 CALCULUS OF VECTOR-VALUED (LT Section 17.5) 622
FUNCTIONS (LT Chapter 14) 250 Chapter Review Exercises 636
13.1 Vector-Valued Functions (LT Section 14.1) 250
13.2 Calculus of Vector-Valued Functions Chapter 17 FUNDAMENTAL THEOREMS OF
(LT Section 14.2) 261 VECTOR ANALYSIS
13.3 Arc Length and Speed (LT Section 14.3) 273 (LT Chapter 18) 649
13.4 Curvature (LT Section 14.4) 282
13.5 Motion in Three-Space (LT Section 14.5) 303 17.1 Green’s Theorem (LT Section 18.1) 649
13.6 Planetary Motion According to Kepler and Newton 17.2 Stokes’ Theorem (LT Section 18.2) 665
(LT Section 14.6) 319 17.3 Divergence Theorem (LT Section 18.3) 678
Chapter Review Exercises 326 Chapter Review Exercises 693
,10 INFINITE SERIES
10.1 Sequences(LT Section 11.1)
Preliminary Questions
1. What is a4 for the sequence an = n2 — n?
SOLUTION Substituting n = 4 in the expression for an gives
a4 = 42 — 4 = 12.
2. Which of the following sequences converge to zero?
n2 µ ¶
(b) 2 n —1 n
(a) n2 + 1 (c)
2
SOLUTION
(a) This sequence does not converge to zero:
n2 x2 1 = 1
lim 2
= lim 2
= lim 1 = 1.
n→∞ n +1 x→∞ x +1 x→∞ 1+ 1+0
x2
(b) This sequence does not converge to zero: this is a geometric sequence with r = 2 > 1; hence, the sequence diverges
to ∞.
(c) Recall that if |an| converges to 0, then an must also converge to zero. Here,
µ ¶ µ ¶n
¯ 1 n¯ 1
¯ —2 ¯ = 2 ,
which is a geometric sequence with 0 < r < 1; hence, ( 1 )n converges to zero. It therefore follows that (— 1 )n converges
2 2
to zero.
√
3. Let an be the nth decimal approximation to 2. That is, a1 = 1, a2 = 1.4, a3 = 1.41, etc. What is lim an?
n→∞
√
SOLUTION an = 2.
lim
n→∞
4. Which of the following sequences is defined recursively?
√ ,
(a) an = 4 + n (b) bn = 4 + bn—1
SOLUTION
(a) an can be computed directly, since it depends on n only and not on preceding terms. Therefore an is defined explicitly
and not recursively.
(b) bn is computed in terms of the preceding term bn—1, hence the sequence {bn} is defined recursively.
5. Theorem 5 says that every convergent sequence is bounded. Determine if the following statements are true or false
and if false, give a counterexample.
(a) If {an} is bounded, then it converges.
(b) If {an} is not bounded, then it diverges.
(c) If {an} diverges, then it is not bounded.
SOLUTION
(a) This statement is false. The sequence an = cos πn is bounded since —1 ≤ cos πn ≤ 1 for all n, but it does not
converge: since an = cos nπ = ( —1)n, the terms assume the two values 1 and —1 alternately, hence they do not approach
one value.
(b) By Theorem 5, a converging sequence must be bounded. Therefore, if a sequence is not bounded, it certainly does
not converge.
(c) The statement is false. The sequence an = (—1)n is bounded, but it does not approach one limit.
, 2 C H A P T E R 10 INFINITE SERIES (LT CHAPTER 11)
Exercises
1. Match each sequence with its general term:
a1, a2, a3, a 4 ,... General term
(a) 21 , 32 , 34 , 45 ,. . . (i) cos πn
n!
(b) —1, 1, —1, 1 ,... (ii) n
2
(c) 1, —1, 1, —1 ,... (iii) (—1)n+1
n
(d) 21 , 42 , 68 , 24
16 ... (iv)
n+ 1
SOLUTION
(a) The numerator of each term is the same as the index of the term, and the denominator is one more than the numerator;
hence an = n n+1 , n = 1, 2, 3 ,. . . .
(b) The terms of this sequence are alternating between —1 and 1 so that the positive terms are in the even places. Since
cos πn = 1 for even n and cos πn = —1 for odd n, we have an = cos πn, n = 1, 2 ,. . . .
(c) The terms an are 1 for odd n and —1 for even n. Hence, an = (—1)n+1, n = 1, 2,...
(d) The numerator of each term is n!, and the denominator is 2n; hence, an = n! 2n , n = 1, 2, 3,....
In Exercises 3–12, calculate the first four terms of the sequence, starting with n = 1.
3n
3. cn =
n!
SOLUTION Setting n = 1, 2, 3, 4 in the formula for cn gives
31 3 32 9
c1 = = = 3, c2 = =
,
1! 1 2! 2
33 27 9 34 81 27
c3 = = = , c4 = = = .
3! 6 2 4! 24 8
5. a1 = 2, an+1 = 2a2n— 3
SOLUTION For n = 1, 2, 3 we have:
a2 = a1+1 = 2a21— 3 = 2 · 4 — 3 = 5;
a3 = a2+1 = 2a22— 3 = 2 · 25 — 3 = 47;
a4 = a3+1 = 2a23— 3 = 2 · 2209 — 3 = 4415.
The first four terms of {an} are 2, 5, 47, 4415.
7. bn = 5 + cos πn
SOLUTION For n = 1, 2, 3, 4 we have
b1 = 5 + cos π = 4;
b2 = 5 + cos 2π = 6;
b3 = 5 + cos 3π = 4;
b4 = 5 + cos 4π = 6.
1 1of {bn} are 14, 6, 4, 6.
The first four terms
9. cn = 1
+ + + ··· +
2 3 n
SOLUTION
c1 = 1;
1 3
c2 = 1 + = ;
2 2
1 1 3 1 11
c3 = 1 + + = + = ;
2 3 2 3 6
1 1 1 11 1 25
c4 = 1 + + + = + = .
2 3 4 6 4 12
All Chapters Included
, CONTENTS
Chapter 10 INFINITE SERIES (LT Chapter 11) 1 Chapter 14 DIFFERENTIATION IN SEVERAL
VARIABLES (LT Chapter 15) 336
10.1 Sequences (LT Section 11.1) 1
10.2 Summing an Infinite Series (LT Section 11.2) 13 14.1 Functions of Two or More Variables
10.3 Convergence of Series with Positive Terms (LT Section 15.1) 336
(LT Section 11.3) 23 14.2 Limits and Continuity in Several Variables
10.4 Absolute and Conditional Convergence (LT Section 15.2) 345
(LT Section 11.4) 38 14.3 Partial Derivatives (LT Section 15.3) 351
10.5 The Ratio and Root Tests (LT Section 11.5) 44 14.4 Differentiability and Tangent Planes
10.6 Power Series (LT Section 11.6) 52 (LT Section 15.4) 363
10.7 Taylor Series (LT Section 11.7) 64 14.5 The Gradient and Directional Derivatives
Chapter Review Exercises 81 (LT Section 15.5) 374
14.6 The Chain Rule (LT Section 15.6) 386
14.7 Optimization in Several Variables
Chapter 11 PARAMETRIC EQUATIONS, (LT Section 15.7) 399
POLAR COORDINATES, 14.8 Lagrange Multipliers: Optimizing with a Constraint
AND CONIC SECTIONS (LT Section 15.8) 419
(LT Chapter 12) 96 Chapter Review Exercises 442
11.1 Parametric Equations (LT Section 12.1) 96
Chapter 15 MULTIPLE INTEGRATION
11.2 Arc Length and Speed (LT Section 12.2) 112
11.3 Polar Coordinates (LT Section 12.3) 120
(LT Chapter 16) 458
11.4 Area and Arc Length in Polar Coordinates
(LT Section 12.4) 133
15.1 Integration in Two Variables (LT Section 16.1) 458
15.2 Double Integrals over More General Regions
11.5 Conic Sections (LT Section 12.5) 143
(LT Section 16.2) 471
Chapter Review Exercises 154
15.3 Triple Integrals (LT Section 16.3) 489
15.4 Integration in Polar, Cylindrical, and Spherical
Chapter 12 VECTOR GEOMETRY Coordinates (LT Section 16.4) 502
(LT Chapter 13) 166 15.5 Applications of Multiple Integrals
(LT Section 16.5) 521
12.1 Vectors in the Plane (LT Section 13.1) 166 15.6 Change of Variables (LT Section 16.6) 537
12.2 Vectors in Three Dimensions (LT Section 13.2) 176 Chapter Review Exercises 554
12.3 Dot Product and the Angle between Two Vectors
(LT Section 13.3) 184 Chapter 16 LINE AND SURFACE INTEGRALS
12.4 The Cross Product (LT Section 13.4) 198 (LT Chapter 17) 576
12.5 Planes in Three-Space (LT Section 13.5) 212
12.6 A Survey of Quadric Surfaces (LT Section 13.6) 223 16.1 Vector Fields (LT Section 17.1) 576
12.7 Cylindrical and Spherical Coordinates 16.2 Line Integrals (LT Section 17.2) 581
(LT Section 13.7) 230 16.3 Conservative Vector Fields (LT Section 17.3) 599
Chapter Review Exercises 240 16.4 Parametrized Surfaces and Surface Integrals
(LT Section 17.4) 606
16.5 Surface Integrals of Vector Fields
Chapter 13 CALCULUS OF VECTOR-VALUED (LT Section 17.5) 622
FUNCTIONS (LT Chapter 14) 250 Chapter Review Exercises 636
13.1 Vector-Valued Functions (LT Section 14.1) 250
13.2 Calculus of Vector-Valued Functions Chapter 17 FUNDAMENTAL THEOREMS OF
(LT Section 14.2) 261 VECTOR ANALYSIS
13.3 Arc Length and Speed (LT Section 14.3) 273 (LT Chapter 18) 649
13.4 Curvature (LT Section 14.4) 282
13.5 Motion in Three-Space (LT Section 14.5) 303 17.1 Green’s Theorem (LT Section 18.1) 649
13.6 Planetary Motion According to Kepler and Newton 17.2 Stokes’ Theorem (LT Section 18.2) 665
(LT Section 14.6) 319 17.3 Divergence Theorem (LT Section 18.3) 678
Chapter Review Exercises 326 Chapter Review Exercises 693
,10 INFINITE SERIES
10.1 Sequences(LT Section 11.1)
Preliminary Questions
1. What is a4 for the sequence an = n2 — n?
SOLUTION Substituting n = 4 in the expression for an gives
a4 = 42 — 4 = 12.
2. Which of the following sequences converge to zero?
n2 µ ¶
(b) 2 n —1 n
(a) n2 + 1 (c)
2
SOLUTION
(a) This sequence does not converge to zero:
n2 x2 1 = 1
lim 2
= lim 2
= lim 1 = 1.
n→∞ n +1 x→∞ x +1 x→∞ 1+ 1+0
x2
(b) This sequence does not converge to zero: this is a geometric sequence with r = 2 > 1; hence, the sequence diverges
to ∞.
(c) Recall that if |an| converges to 0, then an must also converge to zero. Here,
µ ¶ µ ¶n
¯ 1 n¯ 1
¯ —2 ¯ = 2 ,
which is a geometric sequence with 0 < r < 1; hence, ( 1 )n converges to zero. It therefore follows that (— 1 )n converges
2 2
to zero.
√
3. Let an be the nth decimal approximation to 2. That is, a1 = 1, a2 = 1.4, a3 = 1.41, etc. What is lim an?
n→∞
√
SOLUTION an = 2.
lim
n→∞
4. Which of the following sequences is defined recursively?
√ ,
(a) an = 4 + n (b) bn = 4 + bn—1
SOLUTION
(a) an can be computed directly, since it depends on n only and not on preceding terms. Therefore an is defined explicitly
and not recursively.
(b) bn is computed in terms of the preceding term bn—1, hence the sequence {bn} is defined recursively.
5. Theorem 5 says that every convergent sequence is bounded. Determine if the following statements are true or false
and if false, give a counterexample.
(a) If {an} is bounded, then it converges.
(b) If {an} is not bounded, then it diverges.
(c) If {an} diverges, then it is not bounded.
SOLUTION
(a) This statement is false. The sequence an = cos πn is bounded since —1 ≤ cos πn ≤ 1 for all n, but it does not
converge: since an = cos nπ = ( —1)n, the terms assume the two values 1 and —1 alternately, hence they do not approach
one value.
(b) By Theorem 5, a converging sequence must be bounded. Therefore, if a sequence is not bounded, it certainly does
not converge.
(c) The statement is false. The sequence an = (—1)n is bounded, but it does not approach one limit.
, 2 C H A P T E R 10 INFINITE SERIES (LT CHAPTER 11)
Exercises
1. Match each sequence with its general term:
a1, a2, a3, a 4 ,... General term
(a) 21 , 32 , 34 , 45 ,. . . (i) cos πn
n!
(b) —1, 1, —1, 1 ,... (ii) n
2
(c) 1, —1, 1, —1 ,... (iii) (—1)n+1
n
(d) 21 , 42 , 68 , 24
16 ... (iv)
n+ 1
SOLUTION
(a) The numerator of each term is the same as the index of the term, and the denominator is one more than the numerator;
hence an = n n+1 , n = 1, 2, 3 ,. . . .
(b) The terms of this sequence are alternating between —1 and 1 so that the positive terms are in the even places. Since
cos πn = 1 for even n and cos πn = —1 for odd n, we have an = cos πn, n = 1, 2 ,. . . .
(c) The terms an are 1 for odd n and —1 for even n. Hence, an = (—1)n+1, n = 1, 2,...
(d) The numerator of each term is n!, and the denominator is 2n; hence, an = n! 2n , n = 1, 2, 3,....
In Exercises 3–12, calculate the first four terms of the sequence, starting with n = 1.
3n
3. cn =
n!
SOLUTION Setting n = 1, 2, 3, 4 in the formula for cn gives
31 3 32 9
c1 = = = 3, c2 = =
,
1! 1 2! 2
33 27 9 34 81 27
c3 = = = , c4 = = = .
3! 6 2 4! 24 8
5. a1 = 2, an+1 = 2a2n— 3
SOLUTION For n = 1, 2, 3 we have:
a2 = a1+1 = 2a21— 3 = 2 · 4 — 3 = 5;
a3 = a2+1 = 2a22— 3 = 2 · 25 — 3 = 47;
a4 = a3+1 = 2a23— 3 = 2 · 2209 — 3 = 4415.
The first four terms of {an} are 2, 5, 47, 4415.
7. bn = 5 + cos πn
SOLUTION For n = 1, 2, 3, 4 we have
b1 = 5 + cos π = 4;
b2 = 5 + cos 2π = 6;
b3 = 5 + cos 3π = 4;
b4 = 5 + cos 4π = 6.
1 1of {bn} are 14, 6, 4, 6.
The first four terms
9. cn = 1
+ + + ··· +
2 3 n
SOLUTION
c1 = 1;
1 3
c2 = 1 + = ;
2 2
1 1 3 1 11
c3 = 1 + + = + = ;
2 3 2 3 6
1 1 1 11 1 25
c4 = 1 + + + = + = .
2 3 4 6 4 12
