Chapter 9: Infinite Series - dysoncentralne / FrontPagedysoncentralne.pbworks.com/w/file/fetch/53187450/Chapter 9 Test... · Chapter 9: Infinite Series 45. Consider the series 2 1

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


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


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


4. Match the sequence with its graph.

61na

n=

+

A)

B)

C)

D)

E)

258 Copyright © Houghton Mifflin Company. All rights reserved.



81n

nan

=+

A)

B)

C)

D)

E)




3!

n

nan

=

A)

B)

C)

D)

E)




4( 1)nna = −

A)

B)

C)

D)

E)




( 1)n

nan−

=

A)

B)

C)

D)

E)



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




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


( ) 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



15. Match the series with the graph of its sequence of partial sums.

0

23

n

n

∞

=

∑

A)

B)

C)

D)

E)




0

15 14 4

n

n

∞

=

−

∑

A)

B)

C)

D)

E)




0

25 124 13

n

n

∞

=

−

∑

A)

B)

C)

D)

E)



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


1

2( 1)( 8)( 10)

n

n n n

∞

=

−+ +∑

A) 1

72 B) 2

7 C) 3

8− D) 1

8 E) 1

90−


0

344

n

n

∞

=

∑

A) 16 B) 8 C) 12 D) 4 E) 3


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


0

86n

n

∞

=∑

A) Diverges B) Converges C) Cannot be determined from the methods in the chapter



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


2

1

n

nne

∞ −

=∑

A) Integral Test inconclusive B) Diverges C) Converges


52

lnn

nn

∞

=∑





2

2lnn n n

∞

=∑


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


3 33 32 2 2 2

1 1 1 112 3 4 5

+ + + + +

A) Diverges B) Converges C) Theorem 9.11 is inconclusive


1.261

1n n

∞

=∑

A) Diverges B) Converges C) Theorem 9.11 is inconclusive


21

14 1n n

∞

= −∑

A) Inconclusive B) Diverges C) Converges


71

2n n n

∞

= ⋅∑

A) Diverges B) Converges C) Inconclusive


1

18n n

∞

=

⋅∑

A) Diverges B) Converges C) Inconclusive




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


7 / 910

1– 9n n

∞

=∑ converges or diverges.

A) B) C) Direct Comparison Test does not apply converges diverges


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



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.



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



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




2

1 8nn

n∞

=∑

A) Converges B) Diverges C) Ratio Test is inconclusive


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


1

9 110 1

n

n

nn

∞

=

+ −

∑

A) Converges B) Diverges C) Root Test is inconclusive


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



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



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− + −




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


( ) 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+ +


( ) 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− +



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− − + − − − + −



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) ( , )−∞ ∞



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]



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

∞

=

∈ −∑




0

1 ( 1)1

n n

nx

x

∞

=

= −+ ∑


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+∞

+=

− +∈∑


0

1 ( 1)1

n n

nx

x

∞

=

= −+ ∑


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

∞

=

− ∈∑ −



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

∞

=∑


( ) 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 − − −

+


( )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

−∞−

=

−−∑



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− + − + − + − =− + − + − + −



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



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



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




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

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Chapter 9: Infinite Series - dysoncentralne / FrontPagedysoncentralne.pbworks.com/w/file/fetch/53187450/Chapter 9 Test... · Chapter 9: Infinite Series 45. Consider the series 2 1