Homework 1 | Due July 11 (Monday)
1
1. Let f B [a, b], and suppose that P and P are partitions of [a, b] with P P . Prove
that U (P , f ) U (P , f ).
2. The method used in Example 6.1.6 (d) can be summarized as the following theorem:
Let f B [a, b] and su
Homework 2 | Due July 15 (Friday)
1
1. Let f, g B [a, b] and c R.
b
(a) Prove that
b
a
b
g
f+
(f + g ), and nd an example that satises
a
a
b
b
f+
a
b
g<
b
a
a
(b) Prove
(cf ) =
a
c
b
f
a
b
f
a
if c < 0,
b
a
b
a
b
a
a
if c 0,
c
b
f
a
if c < 0,
b
g.
a
b
f
a
Homework 3 | Due July 20 (Wednesday)
1
1, x = 0
1
, x = m Q (0, 1]
1. Let f (x) =
n
n
0, x Qc [0, 1]
Prove that f is continuous at all irrational points.
2. Prove, for f B [a, b] and a partition Q = cfw_y0 , y1 , . . . , yK ,
U (P , f ) (K 1)(M m)|P| U (
Homework 4 | Due July 25 (Monday)
1
1. Find the following integrals. They may or may not exist depending on p R.
p
x dx ,
(a)
1
p
x dx ,
(b)
xp dx .
(c)
0
1
0
et tx1 dt for x (0, ). Show that
2. The Gamma function is dened by (x) =
(n + 1) = n! for all
1
Homework 5 | Due July 29 (Friday)
1
1. Complete the following proofs that were skipped in lecture:
(a) In the proof of the ratio test, prove that r > 1 =
an = .
n=1
(b) In the proof of the root test, prove that > 1 =
an = .
n=1
(c) Prove that r limn (an )
Homework 6 | Due August 4 (Thursday)
1
1. Prove the conditional convergence of each of the following series:
n ln n
(1)
(a)
(b)
n
n=1
n=1
sin n
n
2. Suppose lim nan = A = 0. Prove that
n
an diverges.
n=1
3. Determine the convergence/divergence of the foll
Homework 7 | Due August 9 (Tuesday)
1
xn
1. Let fn (x) =
on [0, 1].
1 + xn
(a) Prove that fn converges uniformly to 0 on [0, ] for all (0, 1).
(b) Does fn converge uniformly on [0, 1]? Prove or disprove.
2. Prove that if fn converges uniformly on (a, b) a
MthSc 454, Advanced Calculus II (Real Analysis II)
Summer Session II, 2011
MTWRF 1:152:45
Martin E-004
General information
Instructor:
Email:
Web:
Phone:
Oce:
Oce hours:
Text:
Web:
Dr. Matthew Macauley
macaule@clemson.edu
http:/www.math.clemson.edu/macaul