6 SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM29
Exercise 6.4 (Bolzano-Weierstrass)
Every bounded sequence has a convergent subsequence. Hint: Exercise 5.8
Exercise 6.5
Show that the sequence cfw_esin n 1 has a convergent subsequence.
n=1
30
SEQUENCES
126
SERIES
31 The Ratio Test and the nth Root Test
The integral test is hard to apply when the integrand involves factorials or
complicated expressions. In this section we introduce two tests that can be
used to help determine convergence or divergence of
90
RIEMANN INTEGRALS
21 Classes of Riemann Integrable Functions
In this section we discuss some families of Riemann integrable functions,
namely, monotone and continuous functions.
Exercise 21.1
Let f : [a, b] ! R be an increasing function on [a, b].
(a)
102
RIEMANN INTEGRALS
Exercise 23.11
Let f, g : [a, b] ! R be two Riemann integrable functions.
(a) Show that
1
f g = [(f + g)2 f 2 g 2 ].
2
(b) Prove that f g is Riemann integrable.
24 COMPOSITION OF RIEMANN INTEGRABLE FUNCTIONS AND ITS APPLICATIONS103
2
114
SERIES
Exercise 26.2
Is the series 1 ( 1)n convergent or divergent?
n=1
The following result provides a procedure for testing the divergence of a
series. This is known as the the nth term test for convergence.
Exercise 26.3
P1
Suppose that
i=1 an = L.
138
SERIES OF FUNCTIONS
Exercise 32.26
Give an example of a sequence cfw_fn 1 and a function f such that fn ! f
n=1
2
uniformly but fn does not converge uniformly to f 2 .
Exercise 32.27
Give an example of two sequences cfw_fn 1 and cfw_gn 1 such that fn
78
DERIVATIVES
changes sign between a and b.
(b) Establish the same result for f 0 (b) < < f 0 (a).
(c) Show that the condition g 0 (c) 6= 0 for all c 2 [a, b] leads to a contradiction.
Hint: Exercise 18.7. Conclude that there must be a a < c < b such tha
9 PROPERTIES OF LIMITS
41
Hint: The area of a circular sector with radius r and central angle is given
by the formula 1 r2 .
2
(b) Show that limx!0+ sin x = 0.
(c) Show that limx!0 sin x = 0. Thus, we conclude that limx!0 sin x = 0.
Hint: Recall that the
Properties of Real Numbers
In this chapter we review the important properties of real numbers that are
needed in this course.
1 Basic Properties of Absolute Value
In this section, we introduce the absolute value function and we discuss some
of its propert
3 SEQUENCES AND THEIR CONVERGENCE
17
We next introduce the concept of a bounded sequence. This concept provides
us with a divergence test for sequences. We will see that if a sequence is not
bounded then it is divergent.
Denition 6
A sequence cfw_an 1 is
12 PROPERTIES OF CONTINUOUS FUNCTIONS
53
Practice Problems
Exercise 12.5
Suppose that f : R ! R is continuous such that f (x) = 0 for all x 2 Q.
Prove that f (x) = 0 for all x 2 R. Hint: Exercise 3.21
Exercise 12.6
Consider the function
f (x) =
x if x 2 Q
66
DERIVATIVES
Exercise 16.2
Consider the function
f (x) =
x2 sin
0
1
x
if x 6= 0
if x = 0
Show that f is dierentiable at a = 0. What is f 0 (0)?
Exercise 16.3
Show that f (x) = |x| is not dierentiable at 0.
Exercise 16.4
Find the derivative of f (x) = si
150
SERIES OF FUNCTIONS
Practice Problems
Exercise 35.6
Let f (x) = sin x.
(a) Using successive dierentiation nd a formula for f (n) (0).
(b) Show that
P2n (x) = P2n+1 (x) = x
x3 x5
+
3! 5!
n
( 1)n 2n+1 X
x2k+1
+
x
=
( 1)k
.
(2n + 1)!
(2k + 1)!
k=0
(c) F