<<

. 12
( 14)



>>

Then F (b) ’ F (a) represents the percentage of normally distributed data that
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 =

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+



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

<<

. 12
( 14)



>>