153 Additional Questions
Last updated 17/01/05
The majority of the questions below build on and develop further some
of the questions given on the Question Sheets. There is no solution sheet
though any numerical values that are asked for are listed at the
153 Problem Sheet 5
All questions should be attempted. Those marked with a * must be
handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions marked with a * or *. Questions
marked with a # will be
153 Problem Sheet 4
All questions should be attempted. Those marked with a * must be
handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions marked with a * or *. Questions
marked with a # will be
153 Problem Sheet 6
All questions should be attempted. Those marked with a * must be
handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions marked with a * or *. Questions
marked with a # will be
Section 6 Series in General
We now consider series in which some, and in fact possibly innitely
many, of the terms are negative. Given such a series ar we might rst
r=1
think to examine ar , a series of nonnegative terms to which we could
r=1
apply the
153 Problem Sheet 3
All questions should be attempted. Those marked with a * must be
handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions marked with a * or *. Questions
marked with a # will be
153 Problem Sheet 2
All questions should be attempted. Those marked with a * must be
handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions marked with a * or *. Questions
marked with a # will be
Section 3 Sequences and Limits, Continued.
Lemma 3.6 Let cfw_an nN be a convergent sequence for which an = 0 for all
n N and limit = 0. Then there exists N N such that

3 
an 
2
2
for all n N .
In particular this result ensures that if the limit is
Section 2 Bounds
Denitions Let A R be a set of real numbers. A real number is said to
be a lower bound (written as lbA) for A if
a A, a.
A real number is said to be an upper bound (written as ubA) for A if
a A, a .
Note, we do not insist that either or be
153 Problem Sheet 1
All questions should be attempted. Those marked with a * must be
handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions marked with a * or *. Questions
marked with a # will be
Section 3 Sequences and Limits
Denition A sequence of real numbers is an innite ordered list a1 , a2 , a3 ,
a4 , . where, for each n N, an is a real number. We call an the nth term
of the sequence.
Usually (but not always) the sequences that arise in pra
153 Sequences and Series
Section 0 Introduction
Notation
N is the set of all natural numbers, cfw_1, 2, 3, .,
Z is the set of all integers, cfw_., 3, 2, 1, 0, 1, 2, 3, ., so N is the set of
all positive integers,
N0 is the set of all nonnegative integers
Section 5 Series with nonnegative terms
Theorem 5.1 Let ar be a series with nonnegative terms and let sn
r=1
be the nth partial sum for each n N. Then ar is convergent if, and
r=1
only if, cfw_sn nN is bounded.
Proof Since ar 0 for all r N, then sn+1 s
Section 4 Series
Denition Let cfw_an nN be a sequence of real numbers. The innite sum
a1 + a2 + a3 + .
is called a series. We call an the nth term of the series. We denote a1 +
a2 + a3 + . by ar .
r=1
Examples
r
r=1 (1) r = (1) + 2 + (3) + 4 +
1
1
1
1
1