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Unformatted text preview: CHAPTER 3 Sequences, Series, and Limits; the Economics of Finance If you have done Alevel maths you will have studied Sequences and Series (in particular Arithmetic and Geometric ones) before; if not you will need to work carefully through the first two sections of this chapter. Sequences and series arise in many economic applications, such as the economics of finance and investment . Also, they help you to understand the concept of a limit and the significance of the natural number , e . You will need both of these later. ./ 1. Sequences and Series 1.1. Sequences A sequence is a set of terms (or numbers) arranged in a definite order. Examples 1.1 : Sequences (i) 3 , 7 , 11 , 15 ,... In this sequence each term is obtained by adding 4 to the previous term. So the next term would be 19. (ii) 4 , 9 , 16 , 25 ,... This sequence can be rewritten as 2 2 , 3 2 , 4 2 , 5 2 ,... The next term is 6 2 , or 36. The dots( ... ) indicate that the sequence continues indefinitely it is an infinite sequence . A sequence such as 3 , 6 , 9 , 12 (stopping after a finite number of terms) is a finite sequence . Suppose we write u 1 for the first term of a sequence, u 2 for the second and so on. There may be a formula for u n , the n th term: Examples 1.2 : The n th term of a sequence (i) 4 , 9 , 16 , 25 ,... The formula for the n th term is u n = ( n + 1) 2 . (ii) u n = 2 n + 3. The sequence given by this formula is: 5 , 7 , 9 , 11 ,... (iii) u n = 2 n + n . The sequence is: 3 , 6 , 11 , 20 ,... Or there may be a formula that enables you to work out the terms of a sequence from the preceding one(s), called a recurrence relation : Examples 1.3 : Recurrence Relations (i) Suppose we know that: u n = u n 1 + 7 n and u 1 = 1. Then we can work out that u 2 = 1 + 7 2 = 15, u 3 = 15 + 7 3 = 36, and so on, to find the whole sequence : 1 , 15 , 36 , 64 ,... 37 38 3. SEQUENCES, SERIES AND LIMITS (ii) u n = u n 1 + u n 2 , u 1 = 1, u 2 = 1 The sequence defined by this formula is: 1 , 1 , 2 , 3 , 5 , 8 , 13 ,... 1.2. Series A series is formed when the terms of a sequence are added together. The Greek letter (pronounced sigma) is used to denote the sum of: n X r =1 u r means u 1 + u 2 + + u n Examples 1.4 : Series (i) In the sequence 3 , 6 , 9 , 12 ,... , the sum of the first five terms is the series: 3 + 6 + 9 + 12 + 15. (ii) 6 X r =1 (2 r + 3) = 5 + 7 + 9 + 11 + 13 + 15 (iii) k X r =5 1 r 2 = 1 25 + 1 36 + 1 49 + + 1 k 2 Exercises 3.1 : Sequences and Series (1) (1) Find the next term in each of the following sequences: (a) 2 , 5 , 8 , 11 ,... (b) 0 . 25 , . 75 , 1 . 25 , 1 . 75 , 2 . 25 ,... (c) 5 , 1 , 7 ,... (d) 36 , 18 , 9 , 4 . 5 ,......
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 Spring '10
 Vines
 Economics

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