The Chemistry Maths Book, Second Edition

638 Solutions to exercises

29. 11 − 13 x 1 + 13 x

2

1 − 1 x

3

30. 11 + 112 x 1 + 154 x

2

1 + 1108 x

3

1 + 181 x

4

31. 11 − 120 x 1 + 1160 x

2

1 − 1640 x

3

1 + 11280 x

4

1 − 11024 x

5

32. 811 − 1216 x 1 + 1216 x

2

1 − 196 x

3

1 + 116 x

4

33. 7291 + 11458 x 1 + 11215 x

2

1 + 1540 x

3

1 + 1135 x

4

1

• 118 x

5

1 + 1 x

6

34. (i)

(ii)(a

4

1 + 1 b

4

1 + 1 c

4

) 1 + 1 4(a

3

b 1 + 1 a

3

c 1 + 1 b

3

a 1 + 1 b

3

c 1

• 1 c

3

a 1 + 1 c

3

b) 1 + 1 6(a

2

b

2

1 + 1 a

2

c

2

1 + 1 b

2

c

2

) 1

• 1 12(a

2

bc 1 + 1 b

2

ca 1 + 1 c

2

ab)

35. (i)

(ii)(a

3

1 + 1 b

3

1 + 1 c

3

1 + 1 d

3

) 1 + 1 3(a

2

b 1 + 1 a

2

c 1 + 1 a

2

d 1

• 1 b

2

a 1 + 1 b

2

c 1 + 1 b

2

d 1 + 1 c

2

a 1 + 1 c

2

b 1 + 1 c

2

d 1

• 1 d

2

a 1 + 1 d

2

b 1 + 1 d

2

c) 1 + 1 6(abc 1 + 1 abd 1

• 1 acd 1 + 1 bcd)

36. 
    

    10211 37.

    
    


37. 
    

Section 7.4

41. (i) 11 + 13 x 1 + 19 x

2

1 + 127 x

3

1 + 181 x

4

1 + 1243 x

5

1

• 1729 x

6

1 +1-

(ii)|x| 1 < 13

42. (i) 11 − 15 x

2

1 + 125 x

4

1 − 1125 x

6

1 +1-

(ii)

43. (i)

(ii)|x| 1 < 12

44. 
    

    (i)0.000001000001000001 -

    
    


45. 
    

Section 7.5

46.diverges 47.converges

48.converges for all a

49.no conclusion (see Exercise 50)

50.diverges ifa 1 ≤ 11 , converges ifa 1 > 11

51.diverges

qe

T

v

v

=−













−

−

1

1

θ

1

2 4 8 16 32 64 128

2345 6

−+ − + − + −

xx x x x x

• 
  

||x< 15

n

nnn

2

432

12

2651 ++−













3

4

23

21 2

−

• 
  

++

n

()( )nn

3

1110

6

!

!!!!

=

3

3000

1

3

2100

3

!

!!!!

,

!

!!!!

==,

4

211

12

!

!!!

=

4

400

1

4

310

4

4

220

6

!

!!!

,

!

!!!

,

!

!!!

== =,

Section 7.6

52. 
    

    4 53. 1 54. 1

    
    


53. 
    

    1 56. 0 57.

    
    


54. 
    


55. 
    

    11 − 1 x

    
    

2

1 + 1 x

4

1 − 1 x

6

1 + 1 x

8

1

60. 
    

61. 
    

62. 
    

63. 
    

64. 
    

65. 
    

66. 
    

67. (i)

(ii)

68. (i) (ii) 01 < 1 x 1 < 12


69. (i) (ii)all x


70. (i) (ii)all x


71. (i) (ii) 01 < 1 x 1 ≤ 14

Section 7.7

72. (i)

(ii)2.08008214

(iii)

73. 1 74. 122 75.− 2 76. 122

2 0800832 9 2 0800840

3

.<<.

2

12 288

5

20736

5

248832

23 4

+− + −

xx x x

ln 2

12

2

1

−

−

















=

∑

n

n

n

x

∞

n

nn

x

n

=

∑

−−

0

2

1

2

∞

()( )

()!

π 2

e

x

n

n

n

2

0

2

=

∑

−

!

∞

()

n

n

x

=

∑

−

0

1

∞

()

Bb

ab

RT

a

RT

2

2

2

22

2

=− +

BBb

a

RT

01

==−, 1 ,

BBb

a

RT

Bb

01 2

2

==−,= 1 ,

→→

1

2

0

0

2

mcvv as

Tm

cc

=+

















• 
  

















• 
  









1

2

1

3

4

5

8

0

2

24

v

vv

• 
  













x

xxx x

++++

357 9

2 6 24 120

13

9

2

9

2

27

8

23 4

−+ − +x

xx x

−− − 2 − −

8

3

4

32

5

32

3

2

34

x

x

xx

2

4

3

4

15

8

315

4

2835

2

6101418

x

xx x x

−+ − +

1

3 9 27 81 243

23 4

−+ − +

xx x x

1

2

3

8

5

16

35

128

23 4

++ + +

xx x x

1

39

5

81

10

243

23 4

+− + −

xx x x

3