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Chapter 9: Infinite Series
1. Write the first five terms of the sequence.
an = 34
n −
A) 3 9 27 81 243, , , ,4 16 64 256 1024− −
D) 3 9 27 81 243, , , ,4 16 64 256 1024
− − − − −
B) 3 9 27 81 243, , , ,4 16 64 256 1024
− − − E) None of the above
C) 3 9 27 81 243, , , ,4 16 64 256 1024
2. Write the first five terms of the sequence.
an = 4 17( 1)n
n+ −
A) 17 17 17 17–17, , – , , –2 3 4 5
D) 17 17 17 1717, , , ,
2 3 4 5
B) 17 17 17 1717, – , , – ,2 3 4 5
E) 17 17 17 17–17, – , , , –
2 3 4 5
C) 17 17 17 17–17, – , – , – , –2 3 4 5
3. Write the first five terms of the sequence.
an = 2
5 5–n n
1+
A) 9 19 311, , ,4 9 16
, 95
D) 19 49 911, – , , – ,
4 9 16295
B) 7 13 191, , , ,2 3 4
51
E) 9 19 311, – , , – ,
4 9 1695
C) 19 49 911, , , ,4 9 16
295
Copyright © Houghton Mifflin Company. All rights reserved. 257
Chapter 9: Infinite Series
4. Match the sequence with its graph.
61na
n=
+
A)
B)
C)
D)
E)
258 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
5. Match the sequence with its graph.
81n
nan
=+
A)
B)
C)
D)
E)
Copyright © Houghton Mifflin Company. All rights reserved. 259
Chapter 9: Infinite Series
6. Match the sequence with its graph.
3!
n
nan
=
A)
B)
C)
D)
E)
260 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
7. Match the sequence with its graph.
4( 1)nna = −
A)
B)
C)
D)
E)
Copyright © Houghton Mifflin Company. All rights reserved. 261
Chapter 9: Infinite Series
8. Match the sequence with its graph.
( 1)n
nan−
=
A)
B)
C)
D)
E)
262 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
9. Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit.
an = ( )9ln4
nn
A) Sequence converges to 0 D) Sequence converges to -1 B) Sequence diverges E) Sequence diverges to 1 C) Sequence converges to 1
10. Determine the convergence or divergence of the sequence with the given nth term. If the
sequence converges, find its limit.
an = ( )8ln
6
n
n
A) Sequence diverges to 0 D)Sequence converges to 1
6
B) Sequence diverges E) Sequence converges to 0 C) Sequence converges to 1
11. Determine the convergence or divergence of the sequence with the given nth term. If the
sequence converges, find its limit.
an = 410
n
n
A) Sequence converges to 1 D) Sequence diverges B) Sequence converges to 0 E) Sequence converges to –1 C) Sequence converges to 2
12. Write the first five terms of the sequence of partial sums.
1 1 1 14 9 16 25
+ + + + +1
A) 5 49 205 52691, , , ,6 32 132 3200
D) 7 55 215 531, , , ,
4 36 144 36
B) 1 1 1 11, , , ,4 9 16 25
E) 1 45 50 1051, , , ,
2 32 33 64
C) 5 49 205 52691, , , ,4 36 144 3600
Copyright © Houghton Mifflin Company. All rights reserved. 263
Chapter 9: Infinite Series
13. Write the first five terms of the sequence of partial sums.
9 27 81 243+ – + –2 4 8 16
3 –
A) 9 27 81 2433, – , , – , ,...2 4 8 16
D) 3 21 39 1653, , , ,
2 4 8 16− −
B) 3 21 39 1653, , , ,2 4 8 16
− E) None of the above
C) 3 21 39 1653, , , ,2 4 8 16
14. Write the first five terms of the sequence of partial sums.
( ) 11
–6–7 n
n
∞
−=∑
A) 6 6 6 6–6, , , ,7 49 343 2401− −
D) 38 270–6, – , – ,7 49
3168 12618– , –343 2401
B) 37 264–6, – , – ,7 49
1836 12612– , –343 2401
E) 36 258–6, – , – ,
7 491800 12606– , –343 2401
C) 34 240–6, – , – ,7 49
3240 12594– , –343 2401
264 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
15. Match the series with the graph of its sequence of partial sums.
0
23
n
n
∞
=
∑
A)
B)
C)
D)
E)
Copyright © Houghton Mifflin Company. All rights reserved. 265
Chapter 9: Infinite Series
16. Match the series with the graph of its sequence of partial sums.
0
15 14 4
n
n
∞
=
−
∑
A)
B)
C)
D)
E)
266 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
17. Match the series with the graph of its sequence of partial sums.
0
25 124 13
n
n
∞
=
−
∑
A)
B)
C)
D)
E)
Copyright © Houghton Mifflin Company. All rights reserved. 267
Chapter 9: Infinite Series
18. Find the sum of the convergent series.
1
7( 7)( 9n n n
∞
= + +∑ )
A) 217
144 B) 15
16 C) 28
33 D) 119 E)
144155144
19. Find the sum of the convergent series.
1
2( 1)( 8)( 10)
n
n n n
∞
=
−+ +∑
A) 1
72 B) 2
7 C) 3
8− D) 1
8 E) 1
90−
20. Find the sum of the convergent series.
0
344
n
n
∞
=
∑
A) 16 B) 8 C) 12 D) 4 E) 3
21. Find the sum of the convergent series.
0
910
n
n
∞
=
−
∑
A) 11
19 B) 9
19 C) 10
17 D) 10 E)
191117
22. Determine the convergence or divergence of the series.
–
–31
3 n
n n
∞
=∑
A) Cannot be determined by methods of this chapter. B) Diverges C) Converges
23. Determine the convergence or divergence of the series.
0
86n
n
∞
=∑
A) Diverges B) Converges C) Cannot be determined from the methods in the chapter
268 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
24. Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x.
0 6
n
nn
x∞
=∑
A) 6 , –1 16
xx
< <−
D) 6 1,
6 6x
x16
− < <+
B) 6 , –6 66
xx
< <+
E) 6 , –6 6
6x
x< <
−
C) 6 1,6 6
x 16x
− < <−
25. Find all values of x for which the series converges. For these values of x, write the sum
of the series as a function of x.
0
522
n
n
x∞
=
−
∑
A) 4 ,3 77
xx
− <−
< D) 4 ,3 7
7x
x< <
−
B) 4 ,1 99
xx
< <−
E) Series diverges for all x
C) 4 ,1 99
xx
< <+
26. Use the Integral Test to determine the convergence or divergence of the series.
1
64 2n n
∞
= +∑
A) Converges B) Diverges C) Integral Test inconclusive
27. Use the Integral Test to determine the convergence or divergence of the series.
2
1
n
nne
∞ −
=∑
A) Integral Test inconclusive B) Diverges C) Converges
28. Use the Integral Test to determine the convergence or divergence of the series.
52
lnn
nn
∞
=∑
A) Converges B) Diverges C) Integral Test inconclusive
Copyright © Houghton Mifflin Company. All rights reserved. 269
Chapter 9: Infinite Series
29. Use the Integral Test to determine the convergence or divergence of the series.
2
2lnn n n
∞
=∑
A) Converges B) Diverges C) Integral Test inconclusive
30. Use Theorem 9.11 to determine the convergence or divergence of the series.
71 9
4n n
∞
=∑
A) Theorem 9.11 is inconclusive B) Converges C) Diverges
31. Use Theorem 9.11 to determine the convergence or divergence of the series.
3 33 32 2 2 2
1 1 1 112 3 4 5
+ + + + +
A) Diverges B) Converges C) Theorem 9.11 is inconclusive
32. Use Theorem 9.11 to determine the convergence or divergence of the series.
1.261
1n n
∞
=∑
A) Diverges B) Converges C) Theorem 9.11 is inconclusive
33. Determine the convergence or divergence of the series.
21
14 1n n
∞
= −∑
A) Inconclusive B) Diverges C) Converges
34. Determine the convergence or divergence of the series.
71
2n n n
∞
= ⋅∑
A) Diverges B) Converges C) Inconclusive
35. Determine the convergence or divergence of the series.
1
18n n
∞
=
⋅∑
A) Diverges B) Converges C) Inconclusive
270 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
36. Determine the convergence or divergence of the series.
0
310
n
n
∞
=
∑
A) Inconclusive B) Diverges C) Converges
37. Use the Direct Comparison Test (if possible) to determine whether the series
26
19 + 5n n
∞
=∑ converges or diverges.
A) B) C) Direct Comparison Test does not apply converges diverges
38. Use the Direct Comparison Test (if possible) to determine whether the series
7 / 910
1– 9n n
∞
=∑ converges or diverges.
A) B) C) Direct Comparison Test does not apply converges diverges
39. Use the Direct Comparison Test (if possible) to determine whether the series
1
87 – 4
n
nn
∞
=∑
converges or diverges. A) B) C) Direct Comparison Test does not apply converges diverges
40.
Use the Limit Comparison Test (if possible) to determine whether the series 1
1
34 3
n
nn
+∞
= −∑
converges or diverges. A) B) C) Limit Comparison Test does not apply converges diverges
41.
Use the Limit Comparison Test (if possible) to determine whether the series 6 4
1
25n n
∞
= +∑
converges or diverges. A) B) C) Limit Comparison Test does not apply diverges converges
42. Use the Limit Comparison Test (if possible) to determine whether the series
3
81
6 55 5n
nn n
∞
=
−+ +∑ 6
converges or diverges.
A) B) C) Limit Comparison Test does not apply converges diverges
Copyright © Houghton Mifflin Company. All rights reserved. 271
Chapter 9: Infinite Series
43. Which of the series below should be used in the Limit Comparison Test to determine
whether the series 1
4sin3n n
∞
=
∑ converges or diverges? Does this series converge or
diverge? A)
compare to 1
13n
n
∞
=∑ ; converges
D)compare to 2
1
1n n
∞
=∑ ; converges
B) compare to
1
1cosn n
∞
=
∑ ; diverges E)
compare to 1
1n n
∞
=∑ ; diverges
C) compare to
1
43
n
n
∞
=
∑ ; d iverges
n
Theorem 9.14 (Alternating Series Test): Let 0.a > The alternating series n
∑ and 1( 1)n
nn
a∞
=
− 1
1( 1)n
nn
a∞
+
=
−∑ converge if the following two conditions are met 1. li a = 2. am 0nn→∞ 1 ,n a+ ≤ for all n.
44.
Consider the series 1
( 1)ln( 7)
n
n n
∞
=
−+∑ .
Review the Alternating Series Test to determine which of the following statements is true for the given series. A) Since , the series diverges. lim 0nn
a→∞
≠
B) Since , the Alternating Series Test cannot be applied. lim 0nna
→∞≠
C) Since for some n, the series diverges. 1 ,na a+ > n
nD) Since for some n, the Alternating Series Test cannot be applied. 1 ,na a+ >E) The series converges.
272 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
45. Consider the series
2
1
( 1) sin (4 )n
n
nn
∞
=
−∑ .
Review the Alternating Series Test to determine which of the following statements is true for the given series. A) Since , the series diverges. lim 0nn
a→∞
≠
B) Since , the Alternating Series Test cannot be applied. lim 0nna
→∞≠
C) Since 1na + na≤ cannot be shown to be true for all n, the series diverges. D) Since 1na + na≤ cannot be shown to be true for all n, the Alternating Series Test
cannot be applied. E) Since for some n, the series diverges. 0,na <
46.
Consider the series 8
81
( 1) 1514
n
n
nn
∞
=
− ++∑ .
Review the Alternating Series Test to determine which of the following statements is true for the given series. A) The series converges. B) Since , the series diverges. lim 0nn
a→∞
≠
C) Since 1na + na≤ cannot be shown to be true for all n, the series diverges. D) Since 1na + na≤ cannot be shown to be true for all n, the Alternating Series Test
cannot be applied. E) Since for some n, the series diverges. 0,na <
47. Use the Alternating Series Test (if possible) to determine whether the series
( )2
71
5 919 7
n
n
nn n
∞
=
+−
+ +∑ 5 converges or diverges?
A) B) C) Alternating Series Test cannot be applied converges diverges
48. Determine whether the series
1 /
11
( 1) 167
n n
nn
−∞
+=
− 2
∑ converges absolutely, converges
conditionally, or diverges. A) B) C) converges conditionally diverges converges absolutely
49.
Determine whether the series 8/91
( 1)n
n n
∞
=
−∑ converges absolutely, converges conditionally,
or diverges. A) B) C) diverges converges conditionally converges absolutely
Copyright © Houghton Mifflin Company. All rights reserved. 273
Chapter 9: Infinite Series
50. Determine whether the series
1
( 1) (5 )!6 !
n
n
nn
∞
=
−∑ converges absolutely, converges
conditionally, or diverges. A) converges absolutely B) converges conditionally C) diverges
n
Theorem 9.15 (Alternating Series Remainder): If a convergent series satisfies the condition 1 ,na a+ ≤ then the absolute value of the remainder NR involved in approximating the sum S by is less than (or equal to) the first neglected term. NSThat is, 1.N N NS S R a +− = ≤
51.
The series 1
( 1)5 6
n
n n
∞
=
−+∑ is a convergent series. Use Theorem 9.15 to determine the number
off terms required to approximate the sum of this series with an error less than 0.001.
A) 195 B) 196 C) 197 D) 198 E) 199
52. It can be shown that
1
1
( 1) 4ln3 3
n
nn n
+∞
=
− =
∑ .
According to Theorem 9.15, the partial sum 16
1
( 1)3
n
nn n
+
=
−∑ approximates 43
ln with error
less than how much? A)
( )7
1 0.000065327 3
≈ D)
( )6
1 0.000228626 3
≈
B)
( )7
1 0.000076216 3
≈ E)
( )5
1 0.000823055 3
≈
C)
( )6
1 0.000195967 3
≈
53. Use the Ratio Test to determine the convergence or divergence of the series.
1
49
n
nn
∞
=
∑
A) Diverges B) Converges C) Ratio Test is inconclusive
274 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
54. Use the Ratio Test to determine the convergence or divergence of the series.
2
1 8nn
n∞
=∑
A) Converges B) Diverges C) Ratio Test is inconclusive
55. Use the Ratio Test to determine the convergence or divergence of the series.
1
21
3( 1)2
nn
n n
−∞
=
− ∑
A) Converges B) Ratio Test is inconclusive C) Diverges
56. Use the Root Test to determine the convergence or divergence of the series.
1
109 1
n
n
nn
∞
=
+
∑
A) Converges B) Diverges C) Root Test is inconclusive
57. Use the Root Test to determine the convergence or divergence of the series.
1
9 110 1
n
n
nn
∞
=
+ −
∑
A) Converges B) Diverges C) Root Test is inconclusive
58. Use the Root Test to determine the convergence or divergence of the series.
2
21
4 110 1
n
n
nn
∞
=
+ −
∑
A) Root Test is inconclusive B) Diverges C) Converges
59. Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used.
1
( 1) 25
n
n n
∞
=
−∑
A) Converges; Alternating Series Test D) Diverges; Integral Test B) Converges; Integral Test E) Both A and B C) Diverges; Ratio Test F) Both C and D
Copyright © Houghton Mifflin Company. All rights reserved. 275
Chapter 9: Infinite Series
60. Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used.
–71
3n n
∞
=∑
A) Diverges; p-series D) Converges; Ratio Test B) Converges; p-series E) Both A and C C) Diverges; Integral Test F) Both B and D
61. Determine the convergence or divergence of the series using any appropriate test from
this chapter. Identify the test used.
21
5n n
∞
=∑
A) Diverges; p-series D) Diverges; Integral Test B) Converges; p-series E) Both A and C C) Converges; Ratio Test F) Both B and D
62. Determine the convergence or divergence of the series using any appropriate test from
this chapter. Identify the test used.
1
105n
nn
∞
= +∑
A) Diverges; Ratio Test B) Diverges; Theorem 9.9 (nth Term Test for Divergence) C) Converges; p-series D) Converges; Integral Test E) Both A and B F) Both C and D
63.
The terms of a series ∑ are defined recursively. Determine the convergence or
divergence of the series. Explain your reasoning. 1
nn
a∞
=
1 12 + 15,4 + 4n n
na an+= = a
A) Diverges; Alternating Series Test D) Converges; Ratio Test B) Converges; Integral Test E) Both A and B C) Diverges; Root Test F) Both C and D
276 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
64. Find a first-degree polynomial function P1 whose value and slope agree with the value and slope of f at x = c. What is P1 called?
3
4( ) , 27f x cx
= =
A) 4 4 ( 27)3 243
y x= + + ; Secant line to ( )y f x= at 27x =
B) 4 4 ( 27)3 243
y x= + − ; Secant line to ( )y f x= at 27x =
C) 4 4 ( 27)3 243
y x= + − ; Tangent line to ( )y f x= at 27x =
D) 4 4 ( 27)3 243
y x= − − ; Tangent line to ( )y f x= at 27x =
E) Both B and C
65. Find a first-degree polynomial function P1 whose value and slope agree with the value and slope of f at x = c. What is P1 called?
( ) tan ,3
f x x c π= =
A) – 3 4 –
3x y π=
; Differential of ( )y f x= at
3x π=
B) – 3 4 –
3y x π=
; Tangent line to ( )y f x= at
3x π=
C) – 2 4 –
3y x π=
; Secant line to ( )y f x= at
3x π=
D) – 3 –4 –
3y x π=
; Tangent line to ( )y f x= at
3x π=
E) None of the above
66. Find the Maclaurin polynomial of degree 3 for the function.
5( ) xf x e−= A) 2 325 1251 5
2 6x x x− + − +
D) 2 325 1251 52 6
x x x− − −
B) 2 325 1251 52 6
x x x+ + + E) 2 325 1251 5
2 6x x x− + +
C) 2 325 1251 52 6
x x x− + −
Copyright © Houghton Mifflin Company. All rights reserved. 277
Chapter 9: Infinite Series
67. Find the Maclaurin polynomial of degree 4 for the function.
7( ) xf x e= A) 2 349 343 24011 7
2 4 64x x x+ + + + x
D) 2 349 343 24011 76 12 48
4x x x− + − + x
B) 2 349 343 24011 76 12 48
4x x x+ + + + x E) 2 349 343 24011 7
2 6 244x x x+ + + + x
C) 2 349 343 24011 72 6 24
4x x x− + − + x
68. Find the Maclaurin polynomial of degree 5 for the function.
( ) sin(2 )f x x=
A) 2 42 223 3
x x+ + D) 3 54 42
3 15x x x− +
B) 3 52 223 3
x x x− + E) 3 58 322
3 5x x x− +
C) 3 54 423 15
x x x+ +
69. Find the Maclaurin polynomial of degree 4 for the function.
( ) cos(7 )f x x=
A) 2 449 240112 24
x x+ − D) 2 4343 168071
6 120x x+ −
B) 2 4343 1680716 120
x x− + E) 3 5343 16807
6 120x x x− −
C) 2 449 240112 24
x x− +
278 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
70. Find the fourth degree Maclaurin polynomial for the function.
1( )6
f xx
=+
A) 2 36 36 216 1296 7776 4x x x− + − + x B) 2 31 1 1 1 1
6 36 216 1296 77764x x x− + − + x
C) 2 31 1 1 1 16 36 216 1296 7776
4x x x+ + + + x4
D) 2 36 36 216 1296 7776x x x+ + + + x E) 2 31 1 1 1 1
6 36 216 1296 77764x x x+ − + − x
71. Find the third degree Taylor polynomial centered at c = 4 for the function.
( )f x x=
A) 2 31 1 12 ( 4) ( 4) ( 4)4 64 512
x x x+ − − − + −
B) 2 31 1 12 ( 4) ( 4) ( 4)4 64 512
x x x− − + − − −
C) 2 31 1 12 ( 4) ( 4) ( 4)4 64 512
x x x− − − − − −
D) 2 31 1 12 ( 4) ( 4) ( 4)4 64 512
x x x+ + − + + +
E) 2 31 1 12 ( 4) ( 4) ( 4)4 64 512
x x x− + + + − +
72. Find the fourth degree Taylor polynomial centered at c = 10 for the function.
( ) lnf x x=
A) 2 3ln10 10( 10) 200( 10) 3000( 10) 40000( 10)x x x x+ − − − + − − − 4 B) 2 31 1 1 1ln10 ( 10) ( 10) ( 10) ( 10)
10 200 3000 40000x x x x− − + − − − + − 4
C) 2 31 1 1 1ln10 ( 10) ( 10) ( 10) ( 10)10 200 3000 40000
x x x x− − − − − − − − 4
D) 2 31 1 1 1ln10 ( 10) ( 10) ( 10) ( 10)10 200 3000 40000
x x x x+ − − − + − − − 4
4
E) 2 3ln10 10( 10) 200( 10) 3000( 10) 40000( 10)x x x x− − + − − − + −
Copyright © Houghton Mifflin Company. All rights reserved. 279
Chapter 9: Infinite Series
73. Find the radius of convergence of the power series.
0
( 1)9
n n
nn
x∞
=
−∑
A) 1
9 B) 9 C) 81 D) 1
81 E) ∞
74. Find the radius of convergence of the power series.
2
0
(10 )(2 )!
n
n
xn
∞
=∑
A) 0 B) 10 C) 20 D) 100 E) ∞
75. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)
0 3
n
n
x∞
=
∑
A) [ B) C) )3,3− ( 3,3)− [ ]3,3− D) 1 1,
3 3 −
E) 1 1,3 3
−
76. Find the interval of convergence of the power series. (Be sure to include a check for
convergence at the endpoints of the interval.)
( )0
6(3 )!
n
n
xn
∞
=∑
A) B) (–1,1) C) [–1,1) D) (–6,6) E) ( , )−∞ ∞
1 1,6 6
−
77. Find the interval of convergence of the power series. (Be sure to include a check for
convergence at the endpoints of the interval.)
0
( 1) !( 10)3
n n
nn
n x∞
=
− −∑
A) (–10,10) B) [–10,10] C) {10} D) {0} E) ( , )−∞ ∞
280 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
78. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)
1
11
( 7)7
n
nn
x −∞
−=
−∑
A) B) C) ( 14,14)− (0,7) ( 7,7)− D) (7, E) None of the above 14)
79. Find the interval of convergence of (i) f(x), (ii) ( )f x′ , (iii) ( )f x′′ , and (iv) ( )f x dx∫ of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)
0( )
9
n
n
xf x∞
=
=
∑
A) (i) (–9,9) ; (ii) (–9,9) ; (iii) (–9,9) ; (iv) [–9,9) B) (i) [–9,9] ; (ii) [–9,9) ; (iii) [–9,9) ; (iv) [–9,9) C) (i) (–9,9) ; (ii) (–9,9) ; (iii) (–9,9) ; (iv) (–9,9) D) (i) [–9,9] ; (ii) [–9,9) ; (iii) (–9,9) ; (iv) [–9,9) E) (i) (–9,9) ; (ii) (–9,9) ; (iii) [–9,9) ; (iv) [–9,9)
80. Find the interval of convergence of (i) f(x), (ii) ( )f x′ , (iii) ( )f x′′ , and (iv) ( )f x dx∫ of
the power series. (Be sure to include a check for convergence at the endpoints of the interval.)
1
1
( 1) ( 7)( )n n
n
xf xn
+∞
=
− −=∑
A) (i) (6,8] ; (ii) (6,8) ; (iii) (6,8] ; (iv) [6,8] B) (i) (6,8] ; (ii) (6,8) ; (iii) (6,8) ; (iv) [6,8) C) (i) (6,8] ; (ii) (6,8] ; (iii) (6,8] ; (iv) [6,8] D) (i) (6,8] ; (ii) (6,8) ; (iii) (6,8) ; (iv) [6,8] E) (i) (6,8] ; (ii) (6,8) ; (iii) [6,8] ; (iv) [6,8]
Copyright © Houghton Mifflin Company. All rights reserved. 281
Chapter 9: Infinite Series
81. Find a geometric power series for the function centered at 0, (i) by the technique shown in Examples 1 and 2 and (ii) by long division.
5( )6
f xx
=−
A)
05 , 6
6
n
n
x x∞
=
− <
∑ D)
0
5 , 66 6
n
n
x x∞
=
<
∑
B) ( )
0
5 , 16
n
nx x
∞
=
− <∑ E) None of the above
C) ( )
0
5 6 , 66
n
nx x
∞
=
− <∑
82. Find a power series for the function, centered at c, and determine the interval of
convergence.
7( ) , 81
f x cx
= =+
A) 1
0
( 1) 7 ( 8) , (–7,9)9
nn
nn
x x∞
+=
−− ∈∑
D) 1
10
( 1) 7 ( 8) , (–7,9)9
nn
nn
x x+∞
+=
−− ∈∑
B) 1
10
( 1) 7 ( 8) , (–1,17)9
nn
nn
x x+∞
+=
−− ∈∑
E)
0
( 1) 7 ( 8) , (–7,9)9
nn
nn
x x∞
=
−− ∈∑
C) 1
0
( 1) 7 ( 8) , (–1,17)9
nn
nn
x x∞
+=
−− ∈∑
83. Use the power series
0
1 ( 1)1
n n
nx
x
∞
=
= −+ ∑
to determine a power series, centered at 0, for the function. Identify the interval of convergence.
2
–8( )1
h xx
=−
A) 2
0( 1) 8 , ( 1,1)n n
nx x
∞
=
− ∈∑ −
) )
D)
0( 1) 8 , ( 1,1)n n
nx x
∞
=
− ∈ −∑
B) 2
08 , ( 1,1n
nx x
∞
=
∈ −∑ E)
04 , ( 1,1n
nx x
∞
=
∈ −∑
C)
08 , ( 1,1)n
nx x
∞
=
∈ −∑
282 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
84. Use the power series
0
1 ( 1)1
n n
nx
x
∞
=
= −+ ∑
to determine a power series, centered at 0, for the function. Identify the interval of convergence.
2
3 2
6 3( )( 5) 5
df xx dx x
= = + +
A) 3
0
( 1) 3( )( 1) , (–5,5)5
nn
nn
n n x x∞
+=
− +∈∑
B) 3
0
( 1) 3( 2)( 1) , (–5,5)5
nn
nn
n n x x∞
+=
− + +∈∑
C) 1
30
( 1) 3( 1) , (–5,5)5
nn
nn
n x x+∞
+=
− +∈∑
D) 11
30
( 1) 3( )( 1) , (–5,5)5
nn
nn
n n x x+∞
++
=
− +∈∑
E) 2 2
30
( 1) 3( 1) , (–5,5)5
nn
nn
n x x+∞
+=
− +∈∑
85. Use the power series
0
1 ( 1)1
n n
nx
x
∞
=
= −+ ∑
to determine a power series, centered at 0, for the function. Identify the interval of convergence.
2
1( )36 1
f xx
=+
A) ( )
0( 1) 36 , 1,1n n n
nx x
∞
=
− ∈∑ − D)
( )2
0( 1) 6 , 6,6n n n
nx x
∞
=
− ∈ −∑
B)
0
1 1( 1) 6 , ,6 6
n n n
nx x
∞
=
− ∈ −
∑ E) 2
0
1 1( 1) 36 , ,6 6
n n n
nx x
∞
=
− ∈ −
∑
C) ( )2
0( 1) 36 , 1,1n n n
nx x
∞
=
− ∈∑ −
Copyright © Houghton Mifflin Company. All rights reserved. 283
Chapter 9: Infinite Series
86. Use the definition to find the Taylor series (centered at c) for the function.
5( ) , 0xf x e c= = A) 2
0
5!
nn
nx
n
∞
=∑
D) 2
0
5 ( 1)!
nn n
nx
n
∞
=
−∑
B)
0
5 ( 1)!
nn n
nx
n
∞
=
−∑ E)
2
0
5(2 )!
nn
nx
n
∞
=∑
C)
0
5!
nn
nx
n
∞
=∑
87. Use the definition to find the Taylor series (centered at c) for the function.
( ) sin( ),4
f x x c π= =
A) 2 32 2 2 2– + –2 2 4 2(2!) 4 2(3!) 4
x x xπ π π − − −
+
B) 2 32 2 2 2+ – –2 2 4 2(2!) 4 2(3!) 4
x x xπ π π − − −
+
C) 2 32 2 2 2– – +2 2 4 2(2!) 4 2(3!) 4
x x xπ π π − − −
–
D) 2 32 2 2 2– – –2 2 4 2! 4 3! 4
x x xπ π π − − −
–
E) 2 32 2 2 2+ + +2 2 4 2(2!) 4 2(3!) 4
x x xπ π π − − −
+
88. Use the definition to find the Taylor series (centered at c) for the function.
( )3( ) ln , 1f x x c= =
A) 11
1
( 1) (3)( 1)1
nn
nx
n
−∞−
=
−−
−∑ D) 1
1
( 1) (3)( 1)1
nn
nx
n
−∞
=
−−
−∑
B)
1
( 1) (3)( 1)n
n
nx
n
∞
=
−−∑
E) 1
1
( 1) (3)( 1)n
n
nx
n
−∞
=
−−∑
C) 11
1
( 1) (3)( 1)n
n
nx
n
−∞−
=
−−∑
284 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
89. Use the binomial series to find the Maclaurin series for the function.
9
1( )1
f xx
=−
A) 2 3 22 3
1 (1 9) (1 9)(1 2 9) 1 5 951 19 2!9 3!9 9 81 2187
x x x x x x+ + + ⋅+ + + + = + + + +3
B) 2 3 22 3
1 (1 9) (1 9)(1 2 9) 1 5 951 19 2!9 3!9 9 81 2187
x x x x x x+ + + ⋅− + − + = − + − +3
C) 2 3 22 3
1 (1 9) (1 9)(1 2 9) 1 5 951 19 2!9 3!9 9 81 2187
x x x x x x+ + + ⋅− − − − = − − − −3
D) 2 4 6 2 4 62 3
1 (1 9) (1 9)(1 2 9) 1 5 951 19 2!9 3!9 9 81 2187
x x x x x x+ + + ⋅+ + + + = + + + +
E) 2 4 6 2 4 62 3
1 (1 9) (1 9)(1 2 9) 1 5 951 19 2!9 3!9 9 81 2187
x x x x x x+ + + ⋅− − − − = − − − −
90. Use the binomial series to find the Maclaurin series for the function.
9( ) 1f x x= +
A) 9 18 27 36 45 9 18 27 36 452 3 4 5
1 3 3(5) 3(5)(7) 3(5)(7)(9) 1 3 5 35 631 12 2!2 3!2 4!2 5!2 2 8 16 128 256
x x x x x x x x x x− − − − − − = − − − − − −
B) 9 18 27 36 45 9 18 27 36 452 3 4 5
1 1 3 3(5) 3(5)(7) 1 3 5 35 631 12 2!2 3!2 4!2 5!2 2 8 16 128 256
x x x x x x x x x x− + − + − + = − + − + − +
C) 9 18 27 36 45 9 18 27 36 45
2 3 4 5
1 3 3(5) 3(5)(7) 3(5)(7)(9) 1 3 5 35 631 12 2!2 3!2 4!2 5!2 2 8 16 128 256
x x x x x x x x x x+ + + + + + = + + + + + +
D) 9 18 27 36 45 9 18 27 36 452 3 4 5
1 1 3 3(5) 3(5)(7) 1 3 5 35 631 12 2!2 3!2 4!2 5!2 2 8 16 128 256
x x x x x x x x x x+ − + − + − = + − + − + −
E) 9 18 27 36 45 9 18 27 36 452 3 4 5
1 1 3 3(5) 3(5)(7) 1 3 5 35 631 12 2!2 3!2 4!2 5!2 2 8 16 128 256
x x x x x x x x x x− + − + − + − =− + − + − + −
Copyright © Houghton Mifflin Company. All rights reserved. 285
Chapter 9: Infinite Series
Answer Key
1. B Section: 9.1
2. A Section: 9.1
3. A Section: 9.1
4. C Section: 9.1
5. B Section: 9.1
6. C Section: 9.1
7. D Section: 9.1
8. E Section: 9.1
9. A Section: 9.1
10. E Section: 9.1
11. B Section: 9.1
12. C Section: 9.2
13. D Section: 9.2
14. E Section: 9.2
15. A Section: 9.2
16. B Section: 9.2
17. C Section: 9.2
18. D Section: 9.2
19. E Section: 9.2
20. A Section: 9.2
21. D Section: 9.2
22. A Section: 9.2
286 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
23. B Section: 9.2
24. E Section: 9.2
25. D Section: 9.2
26. B Section: 9.3
27. C Section: 9.3
28. A Section: 9.3
29. B Section: 9.3
30. C Section: 9.3
31. A Section: 9.3
32. B Section: 9.3
33. C Section: 9.3
34. B Section: 9.3
35. A Section: 9.3
36. C Section: 9.3
37. A Section: 9.4
38. B Section: 9.4
39. B Section: 9.4
40. A Section: 9.4
41. A Section: 9.4
42. A Section: 9.4
43. E Section: 9.4
44. E Section: 9.5
45. D Section: 9.5
Copyright © Houghton Mifflin Company. All rights reserved. 287
Chapter 9: Infinite Series
46. B Section: 9.5
47. A Section: 9.5
48. B Section: 9.5
49. A Section: 9.5
50. C Section: 9.5
51. D Section: 9.5
52. A Section: 9.5
53. B Section: 9.6
54. A Section: 9.6
55. C Section: 9.6
56. B Section: 9.6
57. A Section: 9.6
58. C Section: 9.6
59. A Section: 9.6
60. E Section: 9.6
61. B Section: 9.6
62. B Section: 9.6
63. D Section: 9.6
64. D Section: 9.7
65. B Section: 9.7
66. C Section: 9.7
67. E Section: 9.7
68. D Section: 9.7
288 Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 9: Infinite Series
Copyright © Houghton Mifflin Company. All rights reserved. 289
69. C Section: 9.7
70. B Section: 9.7
71. A Section: 9.7
72. D Section: 9.7
73. B Section: 9.8
74. E Section: 9.8
75. B Section: 9.8
76. A Section: 9.8
77. C Section: 9.8
78. E Section: 9.8
79. A Section: 9.8
80. D Section: 9.8
81. D Section: 9.9
82. C Section: 9.9
83. B Section: 9.9
84. B Section: 9.9
85. E Section: 9.9
86. C Section: 9.10
87. B Section: 9.10
88. E Section: 9.10
89. A Section: 9.10
90. D Section: 9.10