Please do not assume that the questions in a real examination will necessarily

# Please do not assume that the questions in a real

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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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