Math 500, Final
(last)
Name (Print): (rst)
Signature:
There are a total of 100+20 points on this 2 hours and 30 minutes exam. This contains 12 pages
(including this cover page) and 10+ 1 problems. Che
HOMEWORK 9
SHUANGLIN SHAO
1. P 175, Ex. 1
Proof. For any x and h,
(x + h)n = xn + nxn1 h +
n n2 2
n
nn
x
h + +
xhn1 +
h.
2
n1
n
Then
n n2
n
n n1
(x + h)n xn
= nxn1 +
x
h + +
xhn2 +
h
.
h
2
n1
n
Hence
HOMEWORK 10
SHUANGLIN SHAO
1. P221. Ex. 1
Proof. If c = 0, for any > 0, there exists =
|c | ,
then for |x y | ,
|f (x) f (y )| = |c| |x y | |c| = .
If c = 0, f 0.
We conclude that f is uniformly conti
MIDTERM REVIEW FOR MATH 500
SHUANGLIN SHAO
1. The limit
Dene limn an = A: For any > 0, there exists N N such that for
any n N ,
|an A| < .
The key in this denition is to realize that the choice of N d
Mathematical Induction
Example: Consider the sum of
n=1
n=2
n=3
n=4
the rst n natural numbers:
sum = 1
sum = 1 + 2 = 3
sum = 1 + 2 + 3 = 6 = (3)(4)/2
sum = 10
= (4)(5)/2
Based on the above, one might
Equivalence relations
Definition: Let X be a set. A relation on X is a subset R of the product X X. If
(x, y) R, then we say that x is related to y.
Youre already familiar with one example of a relati
Decimal expansions
Denition: A decimal expansion is an expression of the form N.a1 a2 a3 an where
(a) N Z, (b) there are a countably innite number of an s, and (c) each an is a decimal
digit - that is
Differentiability and some of its consequences
Denition: A function f : (a, b) R is dierentiable at a point x0 (a, b) if
lim
xx0
f (x) f (x0 )
x x0
exists. If the limit exists for all x0 (a, b) then f
Conditional convergence and rearrangements of infinite series
Denition: A series which converges but does not converge absolutely is said to be conditionally convergent.
n
n=1 (1) /n
Examples: The alt
The Complex Numbers
The set of complex numbers, denoted C, is obtained from the real numbers R by adjoining
the single element i, which has the property that i2 = 1 (so i is not a real number).
The ba
Limits of functions
We now begin the study of functions or maps of the form f : D R, with D R.
Denition: A function from a set D X to another set Y is a rule that associates to each
element x D (calle
Newtons proof of Keplers second law
The general version of Keplers second law states that the regions swept out by the radius
vector of a mass point moving in a central force eld in equal time interva
Cardinal numbers
Denition: A set X is nite if for some n N theres a bijection f : X cfw_1, 2, . . . , n.
We say that the cardinal number of X is n and write Card(X ) = n.
Denition: A set X is innite i
Sets and Mathematical Notation
A set X is a well-dened collection of objects. In this course, the objects will be (mostly)
real numbers. To say that X is well-dened means that there must be an unambig
Problems for Math 500 - Fall, 2009
Absent other instructions from me, homework is due at the beginning of class each Tuesday.
Homework is an absolutely critical part of this course. You may work with
Math 500, Final
(last)
Name (Print): (first)
Signature:
There are a total of 100+20 points on this 2 hours and 30 minutes exam. This contains 12 pages
(including this cover page) and 10+ 1 problems. C
HOMEWORK 7
SHUANGLIN SHAO
1. P143, Ex. 1
b). cn = n. Then
cn+1
= 1.
cn
1
Then the radius of convergence is = 1 = 1. So the series cn xn converges
when |x| < 1. For x = 1, 1, the series diverges becaus
HOMEWORK 6
SHUANGLIN SHAO
1. P129. Ex. 1
b). The series is convergent by Proposition 9.3.8.
d ). The series is convergent by Proposition 9.3.8.
f ). The series is divergent because
lim
n2
= 1.
+1
n n2
Math 500, Final
(last)
Name (Print): (rst)
Signature:
There are a total of 100+20 points on this 2 hours and 30 minutes exam. This contains 12 pages
(including this cover page) and 10+ 1 problems. Che
HOMEWORK 1
SHUANGLIN SHAO
1. P 6. Ex. 1
(a) The series 1 +
sum is
1
4
+
1
16
+ is a geometric series with ratio 1 . Then the
4
1
1
1
4
4
=.
3
(b) The series
1
1
1
1
1
1
1
+
= 1 + ( ) + ( )2 + ( )3
4
HOMEWORK 2
SHUANGLIN SHAO
1. P63, Ex. 1
Proof. We prove it by contradiction. Assume that there exists a rational
number r such that r2 = 3 and r > 0. Since r is a rational number, then
there exists r
HOMEWORK 3
SHUANGLIN SHAO
Abstract. Please send me an email if you nd mistakes. Thanks.
1. P91. Exercise 7.1
For convenience of writing, we will replace n by N in the reasoning below.
(b) cfw_ 2n1 0.
HOMEWORK 4
SHUANGLIN SHAO
1. P108. Ex 1
Proof. 1. The sequence diverges. If the sequence converges, assume that
an A for some A R. Then taking limits on both sides of an+1 =
1 + a2 , we see that
n
A=
HOMEWORK 6
SHUANGLIN SHAO
1. P129. Ex. 1
b). The series is convergent by Proposition 9.3.8.
d ). The series is convergent by Proposition 9.3.8.
f ). The series is divergent because
lim
n2
= 1.
+1
n n2
HOMEWORK 7
SHUANGLIN SHAO
1. P143, Ex. 1
b). cn = n. Then
cn+1
= 1.
cn
1
Then the radius of convergence is = 1 = 1. So the series cn xn converges
when |x| < 1. For x = 1, 1, the series diverges becaus
HOMEWORK 9
SHUANGLIN SHAO
1. P 175, Ex. 1
Proof. For any x and h,
(x + h)n = xn + nxn1 h +
n n2 2
n
nn
x
h + +
xhn1 +
h.
2
n1
n
Then
n n2
n
n n1
(x + h)n xn
= nxn1 +
x
h + +
xhn2 +
h
.
h
2
n1
n
Hence