Please do not assume that the questions in a real examination will necessarily be very
similar to these sample questions. An examination is designed (by definition) to test
you. You
will
get examination questions unlike questions in this guide and each year
there
will
be examination questions different from those in previous years. The whole
point of examining is to see whether you can apply knowledge in familiar
and
unfamiliar
settings. For this reason, it is important that you try as many examples as possible,
from the guide and from the textbooks. This is not so that you can cover any possible
type of question the examiners can think of! It’s so that you get used to confronting
unfamiliar questions, grappling with them, and finally coming up with the solution.
Do not panic if you cannot completely solve an examination question. There are many
marks to be awarded for using the correct approach or method.
1.6
The use of calculators
You will not be permitted to use calculators of any type in the examination. This is not
something that you should panic about: the examiners are interested in assessing that
you understand the key concepts, ideas, methods and techniques, and will set questions
which do not require the use of a calculator.
6

2
Chapter 2
Series of real numbers
Reading
Bryant, Victor.
Yet Another Introduction to Analysis
. Chapter 2.
Binmore, K.G.
Mathematical Analysis: A Straightforward Approach
. Chapter 6.
Brannan, David.
A First Course in Mathematical Analysis
. Chapter 3.
Bartle, R.G. and D.R. Sherbert.
Introduction to Real Analysis
. Chapters 3 and 9.
2.1
Introduction
The first main topic of the course is
series
. This chapter looks at how one can formalise
and deal properly with infinite sums. A key question is whether an infinite sum exists
(that is, whether a series converges).
To understand series, we need to understand
sequences
. We start, therefore, by racing
through some of the results you should know already from
116 Abstract
Mathematics
about sequences. (The discussion of this background material is
deliberately brief: you can find more information in
116 Abstract Mathematics
and
the reading cited.)
2.2
Revision: sequences
Formally, a
sequence
is a function
f
from
N
to
R
. We call
f
(
n
) the
n
th
term
of the
sequence and we often denote the sequence by (
f
(
n
))
∞
n
=1
or simply (
f
(
n
)). Informally a
sequence is an infinite list of real numbers, one for each positive integer; for example,
a
1
, a
2
, a
3
, . . .
We denote it (
a
n
)
∞
n
=1
or (
a
n
) (or indeed, (
a
r
)
,
(
a
i
) etc.). Then we call
a
n
the
n
th
term of
this sequence.
A sequence may be defined by giving an explicit formula for the
n
th
term. For example
the formula
a
n
=
1
n
defines the sequence whose value at the positive integer
n
is
1
n
.
A sequence may also be defined inductively. For instance, we might have
a
1
= 1
,
a
n
+1
=
a
n
2
+
3
2
a
n
(
n
≥
1)
.

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- Summer '19
- Advanced Math