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 an
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 m
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 m
153 Problem Sheet 6
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handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions m
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 serie
153 Problem Sheet 3
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handed in for marking by your supervisor. Hopefully the supervisor will
have time to cover at least the questions m
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 m
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 .
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
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 m
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.
U
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 positi
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 n
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=