lies between a and b. This percentage is given by

b

f (x)dx.

a

Furthermore,

µ+bσ b

1 2

e’x /2 dx.

√

f (x)dx =

2π

µ+aσ a

Proof. The proof of this theorem is omitted.

Exercises 7.1 None available.

7.2 Discontinuities at End Points

De¬nition 7.2.1 (i) Suppose that f is continuous on [a, b) and

lim f (x) = +∞ or ’ ∞.

x’b’

Then, we de¬ne

b x

f (x)dx = lim f (x)dx.

’x’b

a a

If the limit exists, we say that the improper integral converges; otherwise we

say that it diverges.

(ii) Suppose that f is continuous on (a, b] and

lim f (x) = +∞ or ’ ∞.

x’a+

Then we de¬ne,

b b

f (x)dx = lim+ f (x)dx.

x’a

a x

If the limit exists, we say that the improper integral converges; otherwise we

say that it diverges.

Exercises 7.2

1. Suppose that f is continuous on (’∞, ∞) and g (x) = f (x). Then de¬ne

each of the following improper integrals:

300CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

+∞

(a) f (x)dx

a

b

(b) f (x)dx

’∞

+∞

(c) f (x)dx

’∞

2. Suppose that f is continuous on the open interval (a, b) and g (x) = f (x)

on (a, b). De¬ne each of the following improper integrals if f is not

continuous at a or b:

x

f (x)dx, a ¤ x < b

(a)

a

b

f (x)dx, a < x ¤ b

(b)

x

b

(c) f (x)dx

a

+∞

e’x dx = 1

3. Prove that

0

1

π

1

√ dx =

4. Prove that

2

1 ’ x2

0

+∞

1

5. Prove that dx = π

1 + x2

’∞

∞

1 1

6. Prove that dx = , if and only if p > 1.

xp p’1

1

∞

+∞

’x2 2

e’x dx. Use the comparison between

7. Show that e dx = 2

’∞ 0

+∞

2

2

e’x dx exists.

e’x and e’x . Show that

’∞

1

dx

8. Prove that converges if and only if p < 1.

xp

0

7.2. DISCONTINUITIES AT END POINTS 301

+∞

e’x sin(2x)dx.

9. Evaluate

0

+∞

e’4x cos(3x)dx.

10. Evaluate

0

+∞

x2 e’x dx.

11. Evaluate

0

+∞

xe’x dx.

12. Evaluate

0

∞

13. Prove that sin(2x)dx diverges.

0

∞

14. Prove that cos(3x)dx diverges.

0

15. Compute the volume of the solid generated when the area between the

2

graph of y = e’x and the x-axis is rotated about the y-axis.

16. Compute the volume of the solid generated when the area between the

graph of y = e’x , 0 ¤ x < ∞ and the x-axis is rotated

(a) about the x-axis

(b) about the y-axis.

1

17. Let A represent the area bounded by the graph y = , 1 ¤ x < ∞

x

and the x-axis. Let V denote the volume generated when the area A is

rotated about the x-axis.

(a) show that A is +∞

(b) show that V = π

(c) show that the surface area of V is +∞.

(d) Is it possible to ¬ll the volume V with paint and not be able to paint

its surface? Explain.

18. Let A represent the area bounded by the graph of y = e’2x , 0 ¤ x < ∞,

and y = 0.

302CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

(a) Compute the area of A.

(b) Compute the volume generated when A is rotated about the x-axis.

(c) Compute the volume generated when A is rotated about the y-axis.

+∞ +∞

sin x

2

√ dx.

19. Assume that sin(x )dx = (π/8). Compute

x

0 0

+∞ √

2

e’x = π.

20. It is known that

’∞

+∞

2

e’x dx.

(a) Compute

0

+∞

e’x

√ dx.

(b) Compute

x

0

+∞

2

e’4x dx.

(c) Compute

0

De¬nition 7.2.2 Suppose that f (t) is continuous on [0, ∞) and there exist

some constants a > 0, M > 0 and T > 0 such that |f (t)| < M eat for all

t ≥ T . Then we de¬ne the Laplace transform of f (t), denoted L{f (t)}, by

∞

e’st f (t)dt

L{f (t)} =

0

for all s ≥ s0 . In problems 21“34, compute L{f (t)} for the given f (t).

1 if t ≥ 0

22. f (t) = t

21. f (t) =

0 if t < 0

23. f (t) = t2 24. f (t) = t3

25. f (t) = tn , n = 1, 2, 3, · · · 26. f (t) = ebt

27. f (t) = tebt 28. f (t) = tn ebt , n = 1, 2, 3, · · ·

7.2. DISCONTINUITIES AT END POINTS 303

eat ’ ebt aeat ’ bebt

29. f (t) = 30. f (t) =

a’b a’b

1

31. f (t) = sin(bt) 32. f (t) = cos(bt)

b

1

33. f (t) = sinh(bt) 34. f (t) = cosh(bt)

b

De¬nition 7.2.3 For x > 0, we de¬ne the Gamma function “(x) by

+∞

tx’1 e’t dt.

“(x) =

0

+∞

1√

2

e’x =

In problems 35“40 assume that “(x) exists for x > 0 and π.

2

0

√

35. Show that “(1/2) = π 36. Show that “(1) = 1

√

3 π

37. Prove that “(x + 1) = x“(x) 38. Show that “ =

2 2

3√

5

39. Show that “ = π 40. Show that “(n + 1) = n!

2 4

In problems 41“60, evaluate the given improper integrals.

+∞ +∞

dx

’x2

41. 2xe dx 42.

x3/2

0 1

+∞ +∞

dx 4x

43. 44. dx

x5/2 1 + x2

4 1

+∞ +∞

x 4

45. dx 46. dx

(1 + x2 )3/2 x2 ’ 4

1 16

+∞ +∞

1 1

47. dx 48. dx, p > 1

x(ln x)2 x(ln x)p

2 2

304CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

1 2

’x2

ex dx

49. 3xe dx 50.

’∞ ’∞

∞ ∞

2 dx

51. dx 52.

ex + e’x x2 + 9

’∞

0

2 4

dx x

√ √

53. 54. dx

4 ’ x2 16 ’ x2

0 0

+∞

5

dx

x

√

dx 56.

55.

(25 ’ x2 )2/3 x x2 ’ 4

2

0

√

∞

+∞ ’x

e dx

√ dx √

57. 58.

x x(x + 25)

0 0

∞ +∞

e’x 3

x2 e’x dx

59. dx 60.

1 ’ (e’x )2

0 0

7.3

Theorem 7.3.1 (Cauchy Mean Value Theorem) Suppose that two functions

f and g are continuous on the closed interval [a, b], di¬erentiable on the open

interval (a, b) and g (x) = 0 on (a, b). Then there exists at least one number

c such that a < c < b and

f (b) ’ f (a)

f (c)

= .

g(b) ’ g(a)

g (c)

Proof. See the proof of Theorem 4.1.6.

Theorem 7.3.2 Suppose that f and g are continuous and di¬erentiable on

an open interval (a, b) and a < c < b. If f (c) = g(c) = 0, g (x) = 0 on (a, b)

and

f (x)

=L

lim

x’c g (x)

then

f (x)

lim = L.

g(x)

x’c

7.3. 305

Proof. See the proof of Theorem 4.1.7.

Theorem 7.3.3 (L™Hˆpital™s Rule) Let lim represent one of the limits

o

lim, lim , lim , lim , or lim .

’

x’c x’+∞ x’’∞

+

x’c x’c

Suppose that f and g are continuous and di¬erentiable on an open interval

(a, b) except at an interior point c, a < c < b. Suppose further that g (x) = 0

on (a, b), lim f (x) = lim g(x) = 0 or lim f (x) = lim g(x) = +∞ or ’∞. If

f (x)

= L, +∞ or ’ ∞

lim

g (x)

then

f (x) f (x)

lim = lim .

g(x) g (x)

Proof. The proof of this theorem is omitted.

De¬nition 7.3.1 (Extended Arithmetic) For the sake of convenience in deal-

ing with indeterminate forms, we de¬ne the following arithmetic operations

with real numbers, +∞ and ’∞. Let c be a real number and c > 0. Then

we de¬ne

+ ∞ + ∞ = +∞, ’∞ ’ ∞ = ’∞, c(+∞) = +∞, c(’∞) = ’∞

’c

c c

(’c)(+∞) = ’∞, (’c)(’∞) = +∞, = 0, = 0, = 0,

’∞

+∞ +∞

’c

= 0, (+∞)c = +∞, (+∞)’c = 0, (+∞)(+∞) = +∞, (+∞)(’∞) = ’∞,

’∞

(’∞)(’∞) = +∞.

De¬nition 7.3.2 The following operations are indeterminate:

0 +∞ +∞ ’∞ ’∞

, ∞ ’ ∞, 0 · ∞, 00 , 1∞ , ∞0 .

, , ,

0 +∞ ’∞ ’∞ +∞

Remark 23 The L™Hˆpital™s Rule can be applied directly to the 0 and ±∞ 0

o ±∞

±∞

0

forms. The forms ∞ ’ ∞ and 0 · ∞ can be changed to the 0 or ±∞ by

using arithmetic operations. For the 00 and 1∞ forms we use the following

procedure:

ln(f (x))

lim (1/g(x))

g(x) g(x) ln(f (x))

.

lim(f (x)) = lim e =e

It is best to study a lot of examples and work problems.

306CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

Exercises 7.3

1. Prove the Theorem of the Mean: Suppose that a function f is continuous

on a closed and bounded interval [a, b] and f exists on the open interval

(a, b). Then there exists at least one number c such that a < c < b and

f (b) ’ f (a)

(2) f (b) = f (a) + f (c)(b ’ a).

= f (c)

(1)

b’a

2. Prove the Generalized Theorem of the Mean: Suppose that f and g are

continuous on a closed and bounded interval [a, b] and f and g exist

on the open interval (a, b) and g (x) = 0 for any x in (a, b). Then there

exists some c such that a < c < b and

f (b) ’ f (a) f (c)

= .

g(b) ’ g(a) g (c)

3. Prove the following theorem known as l™Hˆpital™s Rule: Suppose that f

o

and g are di¬erentiable functions, except possibly at a, such that

f (x)

= L.

lim f (x) = 0, lim g(x) = 0, and lim

g(x)

x’a x’a x’a

Then

f (x) f (x)

lim = lim = L.

g(x) x’a g (x)

x’a

4. Prove the following theorem known as an alternate form of l™Hˆpital™s

o

Rule: Suppose that f and g are di¬erentiable functions, except possibly

at a, such that

f (x)

lim f (x) = ∞, lim g(x) = ∞, = L.

and lim

g (x)

x’a x’a x’a

Then

f (x) f (x)

lim = lim = L.

g(x) x’a g (x)

x’a

7.3. 307

5. Prove that if f and g exist and

f (x)

lim f (x) = 0, lim g(x) = 0, and lim = L,

g (x)

x’+∞ x’+∞ x’+∞

then

f (x)

lim = L.

g(x)

x’+∞

6. Prove that if f and g exist and

f (x)

lim f (x) = 0, lim g(0) = 0, and lim = L,

x’’∞ g (x)

x’’∞ x’+∞

then

f (x)

lim = L.

g(x)

x’’∞

7. Prove that if f and g exist and

f (x)

lim f (x) = ∞, lim g(x) = ∞, and lim = L,

g (x)

x’+∞ x’+∞ x’+∞

then

f (x)

lim = L.

g(x)

x’+∞

8. Prove that if f and g exist and

f (x)

lim f (x) = ∞, lim g(x) = ∞, and lim = L,

g (x)

x’’∞ x’’∞ x’’∞

then

f (x)

lim = L.

g(x)

x’+∞

9. Suppose that f and f exist in an open interval (a, b) containing c. Then

prove that

f (c + h) ’ 2f (c) + f (c ’ h)

lim = f (c).

h2

h’0

308CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

10. Suppose that f is continuous in an open interval (a, b) containing c.

Then prove that

f (c + h) ’ f (c ’ h)

lim = f (c).

2h

h’0

11. Suppose that f (x) and g(x) are two polynomials such that

f (x) = a0 xn + a1 xn’1 + · · · + an’1 x + an , a0 = 0,

g(x) = b0 xm + b1 xm’1 + · · · + bm’1 x + bm , b0 = 0.

Then prove that

±

0 if m > n

f (x)

= +∞ or ’ ∞ if m < n

lim

g(x)

x’+∞

a0 /b0 if m = n

12. Suppose that f and g are di¬erentiable functions, except possibly at c,

and

lim f (x) = 0, lim g(x) = 0 and lim g(x) ln(f (x)) = L.

x’c x’c x’c

Then prove that

lim (f (x))g(x) = eL .

x’c

13. Suppose that f and g are di¬erentiable functions, except possibly at c,

and

lim f (x) = +∞, lim g(x) = 0 and lim g(x) ln(f (x)) = L.

x’c x’c x’c

Then prove that

lim (f (x))g(x) = eL .

x’c

14. Suppose that f and g are di¬erentiable functions, except possibly at c,

and

lim f (x) = 1, lim g(x) = +∞ and lim g(x) ln(f (x)) = L.

x’c x’c x’c

Then prove that

lim (f (x))g(x) = eL .

x’c

7.3. 309

15. Suppose that f and g are di¬erentiable functions, except possibly at c,

and

f (x)

lim f (x) = 0, lim g(x) = +∞ and lim = L.

(1/g(x))

x’c x’c x’c

Then prove that

lim f (x)g(x) = L.

x’c

1

16. Prove that lim (1 + x) x = e.

x’0

1

1

17. Prove that lim (1 ’ x) x = .

e

x’0

xn

18. Prove that lim = 0 for each natural number n.

ex

x’+∞

sin x ’ x

19. Prove that lim = 0.

x sin x

+x’0

π

’ x tan x = 1.

20. Prove that lim

2

π

x’ 2

In problems 21“50 evaluate each of the limits.

sin(x2 ) 1 ’ cos x2

21. lim 22. lim

x2 x2

x’0 x’0

sin(ax) tan(mx)

23. lim 24. lim

sin(bx) tan(nx)

x’0 x’0

e3x ’ 1

26. lim (1 + 2x)3/x

25. lim

x

x’0 x’0

ex+h ’ ex

ln(x + h) ’ ln(x)

27. lim 28. lim

h h

h’0 h’0

ln(100 + x)

29. lim (1 + mx)n/x 30. lim

x

x’0 x’∞

310CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

31. lim (1 + sin mx)n/x lim (sin x)x

32.

x’0 +

x’0

x4 ’ 2x3 + 10

sin x

33. lim (x) 34. lim

x’∞ 3x4 + 2x3 ’ 7x + 1

x’0+

2π

35. lim tan(2x) ln(x) 36. lim x sin

x

x’+∞

+

x’0

x

3 + 2x

x 2/x

37. lim (x + e ) 38. lim

4 + 2x

x’0 x’∞

39. lim (1 + sin mx)n/x lim (x)sin(3x)

40.

x’0 +

x’0

1 cos 4x

lim (e3x ’ 1)2/ ln x ’

41. 42. lim

x2 x2

x’0

+

x’0

cot(ax) ln x

43. lim 44. lim

cot(bx) x

x’+∞

+

x’0

x 1 2

’

45. lim 46. lim

ln x x ln x

+ +

x’0 x’0

2x + 3 sin x

lim x(b1/x ’ 1), b > 0, b = 1

47. lim 48.

4x + 2 sin x

x’+∞ x’+∞

bx+h ’ bx logb (x + h) ’ logb x

49. lim , b > 0, b = 1 50. lim , b > 0, b = 1

h h

h’0 h’0

(ex ’ 1) sin x x+1

51. lim 52. lim x ln

x’0 cos x ’ cos2 x x’1

x’+∞

2x ’ 3x6 + x7

sin 5x

53. lim 54. lim

(1 ’ x)3

x’0+ 1 ’ cos 4x x’1

ex + 1 tan x ’ sin x

x

55. lim e ln 56. lim

ex x3

x’+∞ x’0

7.3. 311

x3 sin 2x 5x ’ 3x

57. lim 58. lim

x’0 (1 ’ cos x)2 x2

x’0

arctan x ’ x

1 1+x

59. lim ln 60. lim

x3

1’x

x

x’0 x’0

ln(1 + xe2x )

sin(π cos x)

61. lim 62. lim

x2

x sin x

x’0 x’+∞

(ln x)n x + e2x

1

√ ln

, n = 1, 2, · · ·

63. lim 64. lim

x x

x

x’+∞ x’+∞

ln x ln(tan 3x)

65. lim 66. lim

(1 + x3 )1/2 ln(tan 4x)

x’+∞ +

x’0

1/x2

sin x

lim (1 ’ 3’x )’2x

67. 68. lim

x

x’0

+

x’0

x2

3

lim (e’x + e’2x )1/x

69. 70. lim cos

x

x’+∞ x’+∞

x2

x

1 1

71. lim ln 72. lim 1+

x 2x

x’+∞

x’0+

3x+ln x

1 1 1

’

73. lim 1+ 74. lim

2x x sin 2x

x’+∞ x’0

√ 1 1

x2 + b 2 ’ x ’2

75. lim x 76. lim

x sin x x

x’+∞ x’0

5 1 1

1

’2 ’ ln

78. lim

77. lim

x’2 x +x’6 x x

x’2 +

x’0

1 1 1

cot x ’ ’

79. lim 80. lim

x2 tan2 x

x

x’0 x’0

312CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

e’x x ’ sin x

1

’x

81. lim 82. lim

e ’1

x x

x’0 x’∞

1

x2 sin x 1

83. lim 84. lim x sin

sin x x

x’0 x’∞

e ’ (1 + x)1/x ln(ln x)

85. lim 86. lim

ln(x ’ ln x)

x

x’0 x’+∞

x

1 1 1 ln t

’

87. lim 88. lim dt

x2 x ln x x 1+t

x’+∞

+

x’0 1

x

1

x

sin2 x dx

lim (ln(1 + e ) ’ x)

89. 90. lim

x2

x’+∞ x’+∞ 0

91. Suppose that f is de¬ned and di¬erentiable in an open interval (a, b).

Suppose that a < c < b and f (c) exists. Prove that

f (x) ’ f (c) ’ (x ’ c)f (c)

f (c) = lim .

((x ’ c)2 /2!)

x’c

92. Suppose that f is de¬ned and f , f , · · · , f (n’1) exist in an open interval

(a, b). Also, suppose that a < c < b and f (n) (c) exists

(a) Prove that

(x’c)n’1 n’1

f (x) ’ f (c) ’ (x ’ c)f (c) ’ · · · ’ f (c)

(n’1)!

f (n) (c) = lim .

(x’c)n

x’c

n!

(b) Show that there is a function En (x) de¬ned on (a, b), except possibly

at c, such that

(x ’ c)n’1 (n’1)

f (x) = f (c) + (x ’ c)f (c) + · · · + f (x)

(n ’ 1)!

(x ’ c)n (n) (x ’ c)n En (x)

+ f (c) + En (x)

n! n!

7.3. 313

and lim En (x) = 0. Find E2 (x) if c = 0 and

n’c

1

x4 sin , x=0

x

f (x) =

0 , x=0

(c) If f (c) = · · · = f (n’1) (c) = 0, n is even, and f has a relative mini-

mum at x = c, then show that f (n) (c) ≥ 0. What can be said if f has

a relative maximum at c? What are the su¬cient conditions for a rel-

ative maximum or minimum at c when f (c) = · · · = f (n’1) (c) = 0?

What can be said if n is odd and f (c) = · · · = f (n’1) (c) = 0 but

f (n) (c) = 0.

93. Suppose that f and g are de¬ned, have derivatives of order 1, 2, · · · , n’1

in an open interval (a, b), a < c < b, f (n) (c) and g (n) (c) exist and g (n) (c) =

0. Prove that if f and g, as well as their ¬rst n ’ 1 derivatives are 0,

then

f (n) (c)

f (x)

= (n) .

lim

x’c g(x) g (c)

Evaluate the following limits:

π

1

x2 sin x cos cos x

2

94. lim 95. lim

sin2 x

x

x’0 x’0

1

96. lim x( 1’x ) lim x(ln(x))n , n = 1, 2, 3, · · ·

97.

x’1 +

x’0

x3/2 ln x

xx ’ x

99. lim

98. lim

(1 + x4 )1/2

x’1+ 1 ’ x + ln x x’+∞

1 + ex

n

, n = 1, 2, · · ·

100. lim x ln

ex

x’+∞

x ’t2

x e dx

0

101. lim

1 ’ e’x2

x’0

314CHAPTER 7. IMPROPER INTEGRALS AND INDETERMINATE FORMS

7.4 Improper Integrals

1. Suppose that f is continuous on (’∞, ∞) and g (x) = f (x). Then de¬ne

each of the following improper integrals:

Chapter 8

In¬nite Series

8.1 Sequences

De¬nition 8.1.1 An in¬nite sequence (or sequence) is a function, say f ,

whose domain is the set of all integers greater than or equal to some integer

m. If n is an integer greater than or equal to m and f (n) = an , then we

express the sequence by writing its range in any of the following ways:

1. f (m), f (m + 1), f (m + 2), . . .

2. am , am+1 , am+2 , . . .

3. {f (n) : n ≥ m}

4. {f (n)}∞

n=m

5. {an }∞

n=m

De¬nition 8.1.2 A sequence {an }∞ is said to converge to a real number

n=m

L (or has limit L) if for each > 0 there exists some positive integer M such

that |an ’ L| < whenever n ≥ M . We write,

lim an = L or an ’ L as n ’ ∞.

n’∞

If the sequence does not converge to a ¬nite number L, we say that it diverges.

315

316 CHAPTER 8. INFINITE SERIES

Theorem 8.1.1 Suppose that c is a positive real number, {an }∞ and {bn }∞

n=m n=m

are convergent sequences. Then

(i) lim (can ) = c lim an

n’∞ n’∞

(ii) lim (an + bn ) = lim an + lim bn

n’∞ n’∞ n’∞

(iii) lim (an ’ bn ) = lim an ’ lim bn

n’∞ n’∞ n’∞

(iv) lim (an bn ) = lim an lim bn

n’∞ n’∞ n’∞

an limn’∞ an

(v) lim = , if lim bn = 0.

bn limn’∞ bn

n’∞ n’∞

c

(vi) lim (an )c = lim an

n’∞ n’∞

(vii) lim (ean ) = elimn’∞ an

n’∞

(viii) Suppose that an ¤ bn ¤ cn for all n ≥ m and

lim an = lim cn = L.

n’∞ n’∞

Then

lim bn = L.

n’∞

Proof. Suppose that {an }∞ converges to a and {bn }∞ converges to b.

n=m n=m

Let 1 > 0 be given. Then there exist natural numbers N and M such that

|an ’ a| < if n ≥ N, (1)

1

|bn ’ b| < if n ≥ M. (2)

1

and n ≥ N + M . Then

Part (i) Let > 0 be given and c = 0. Let =

1

2|c|

by the inequalities (1) and (2), we get

|can ’ ca| = |c| |an ’ a|

< |c| 1

<.

8.1. SEQUENCES 317

This completes the proof of Part (i).

. Let m ≥ N + M . Then by the

Part (ii) Let > 0 be given and =

1

2

inequalities (1) and (2), we get

|(an + bn ) ’ (a + b)| = |(an ’ a) + (bn ’ b)|

¤ |an ’ a| + |bn ’ b|

< 1+ 1

=.

This completes the proof of Part (ii).

Part (iii)

lim (an ’ bn ) = lim (an + (’1)bn )

n’∞ n’∞

= lim an + lim [(’1)bn ] (by Part (ii))

n’∞ n’∞

= lim an + (’1) lim bn (by Part (i))

n’∞ n’∞

= a + (’1)b

= a ’ b.

. If n ≥ N + M ,

Part (iv) Let > 0 be given and = min 1,

1

1 + |a| + |b|

then by the inequalities (1) and (2) we have

|an bn ’ ab| = |[(an ’ a) + a][(bn ’ b) + b] ’ ab|

= |(an ’ a)(bn ’ b) + (an ’ a)b + a(bn ’ b|

¤ |an ’ a| |bn ’ b| + |b| |an ’ a| + |a| |bn ’ b|

< 2 + |b| 1 + |a| 1

1

= 1 ( 1 + |b| + |a|)

¤ 1 (1 + |b| + |a|)

¤.

Part (v) First we assume that b > 0 and prove that

1 1

lim =.

bn b

n’∞

318 CHAPTER 8. INFINITE SERIES

1

b and using inequality (2) for n ≥ M , we get

By taking =

1

2

1 1 1

|bn ’ b| < b, ’ b < bn ’ b < b,

2 2 2

1 3 2 1 2

b < bn < b, 0 < < <.

2 2 3b bn b

Then, for n ≥ M , we get

b ’ bn

1 1

’ =

b ’ nb

bn b

11

= |bn ’ b| · ·

b bn

2

< |bn ’ b| · 2 . (3)

b

b b2

Let > 0 be given. Choose = min , . There exists some natural

2

22

number N such that if n ≥ N , then

|bn ’ b| < 2. (4)

If n ≥ N + M , then the inequalities (3) and (4) imply that

2

1 1

< |bn ’ b| 2

’

bn b b

2

<22

b

¤.

It follows that

1

1

=

lim

bn b

n’∞

an 1

= lim (an ) · lim

lim

bn bn

n’∞ n’∞ n’∞

1

=a·

b

a

=.

b

8.1. SEQUENCES 319

If b < 0, then

an 1

= lim (’an ) · lim

lim

’bn

bn

n’∞ n’∞ n’∞

1

= (’a)

’b

a

=.

b

This completes the proof of Part (v).

Part (vi) Since f (x) = xc is a continuous function,

c

c

= ac .

lim (an ) = lim an

n’∞ n’∞

Part (vii) Since f (x) = ex is a continuous function,

lim ean = elimn’∞ an

= ea .

n’∞

Part (viii) Suppose that an ¤ bn ¤ cn for all n ≥ m and

lim an = L = lim cn = L.

n’∞ n’∞

Let > 0 be given. Then there exists natural numbers N and M such that

’

< an ’ L < for n ≥ N,

|an ’ L| < ,

2 2 2

’

|cn ’ L| < , < cn ’ L < for n ≥ M.

2 2 2

If n ≥ N + M , then n > N and n > M and, hence,

’ < an ’ L ¤ bn ’ L ¤ cn ’ L < .

2 2

It follows that

lim bn = L.

n’∞

This completes the proof of this theorem.

320 CHAPTER 8. INFINITE SERIES

8.2 Monotone Sequences

De¬nition 8.2.1 Let {tn }∞ be a given sequence. Then {tn }∞ is said

n=m n=m

to be

(a) increasing if tn < tn+1 for all n ≥ m;

(b) decreasing if tn+1 < tn for all n ≥ m;

(c) nondecreasing if tn ¤ tn+1 for all n ≥ m;

(d) nonincreasing if tn+1 ¤ tn for all n ≥ m;

(e) bounded if a ¤ tn ¤ b for some constants a and b and all n ≥ m;

(f) monotone if {tn }∞ is increasing, decreasing, nondecreasing or nonin-

n=m

creasing.

(g) a Cauchy sequence if for each > 0 there exists some M such that

|an1 ’ an2 | < whenever n1 ≥ M and n2 ≥ M .

Theorem 8.2.1 (a) A monotone sequence converges to some real number if

and only if it is a bounded sequence.

(b) A sequence is convergent if and only if it is a Cauchy sequence.

Proof.

Part (a) Suppose that an ¤ an+1 ¤ B for all n ≥ M and some B. Let L be

the least upper bound of the sequence {an }∞ . Let > 0 be given. Then

n=m

there exists some natural number N such that

L ’ < aN ¤ L.

Then for each n ≥ N , we have

L ’ < aN ¤ an ¤ L.

By de¬nition {an }∞ converges to L.

n=m

Similarly, suppose that B ¤ an+1 ¤ an for all n ≥ M . Let L be the

greatest lower bound of {an }∞ . Then {an }∞ converges to L. It follows

n=m n=m

that a bounded monotone sequence converges. Conversely, suppose that a

8.2. MONOTONE SEQUENCES 321

monotone sequence {an }∞ converges to L. Let = 1. Then there exists

n=m

some natural number N such that if n ≥ N , then

|an ’ L| <

’ < an ’ L <

L ’ < an < L + .

The set {an : m ¤ n ¤ N } is bounded and the set {an : n ≥ N } is bounded.

It follows that {an }∞ is bounded. This completes the proof of Part (a) of

n=m

the theorem.

Part (b) First, let us suppose that {an }∞ converges to L. Let > 0 be

n=m

given. Then > 0 and hence there exists some natural number N such that

2

for all natural numbers p ≥ N and q ≥ N , we have

|ap ’ L| < and |aq ’ L| <

2 2

|ap ’ aq | = |(ap ’ L) + (L + aq )|

¤ |ap ’ L| + |a1 ’ L|

< +

2 2

=.

It follows that {an }∞ is a Cauchy sequence.

n=m

Next, we suppose that {an }∞ is a Cauchy sequence. Let S = {an : m ¤

n=m

n < ∞}. Suppose > 0. Then there exists some natural number N such

that for all p ≥ 1

|aN +p ’ aN | < , aN ’ < aN +p < aN + (1)

2 2 2

It follows that S is a bounded set. If S is an in¬nite set, then S has some

limit point q and some subsequence {ank }∞ of {an }∞ that converges to

n=m

k=1

q. Since > 0, there exists some natural number M such that for all k ≥ M ,

we have

|ank ’ q| < (2)

2

322 CHAPTER 8. INFINITE SERIES

Also, for all k ≥ N + M , we get nk ≥ k ≥ N + M and

|ak ’ q| = |ak ’ ank + ank ’ q|

¤ |ank ’ ak | + |ank ’ q|

< +2 (by (1) and (2))

2

=.

It follows that the sequence {an }∞ converges to q. If S is a ¬nite set, then

n=m

some ak is repeated in¬nite number of times and hence some subsequences of

{an }∞ converges to ak . By the preceding argument {an }∞ also converges

n=m n=m

to ak . This completes the proof of this theorem.

Theorem 8.2.2 Let {f (n)}∞ be a sequence where f is a di¬erentiable

n=m

function de¬ned for all real numbers x ≥ m. Then the sequence {f (n)}∞

n=m