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**Unformatted text preview: **A. Alaca MATH 1005 Winter 2010 2 INFINITE SEQUENCES AND SERIES A sequence is an ordered list having a first element but no last element: a 1 , a 2 , a 3 ,...,a n ,... a 1 is the first term, a 2 is the second term, a n is the n th term or general term. Each term a n of an infinite sequence has a successor a n +1 . An infinite sequence is a function whose domain is the set of positive integers N : f : N → R , f ( n ) = a n . Notation: The sequence { a 1 , a 2 , a 3 ,...,a n ,... } is denoted by { a n } or { a n } ∞ 1 . Examples: • { a n } = { 1 , 4 , 9 , 16 ,... } • { a n } ∞ n =1 = { 2 n n + 1 } = { 1 , 4 3 , 6 4 , 8 5 ,... } • { 1 2 n } ∞ n =0 = { 1 , 1 2 , 1 4 , 1 8 ,..., 1 2 n ,... } • { √ n − 7 } ∞ n =7 = { , 1 , √ 2 , √ 3 , 2 ,..., √ n − 7 ,... } • f 1 = 1 , f 2 = 1 , f n = f n- 1 + f n- 2 , n ≥ 3 The last sequence is called the Fibonacci sequence. First few terms are { 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 ,... } . A sequence can be specified in three ways: (i) listing the first few terms followed by ··· if the pattern is obvious, (ii) providing a formula for the general term a n as a function of n , (iii) providing a formula for calculating the term a n as a function of earlier terms a 1 , a 2 , ..., a n- 1 and specify enough of the beginning terms so the proces of computing higher terms can begin. But there are some sequences that do not have a simple defining equation. Example: If a n is the digit in the n th decimal place of the number e , then { a n } is a sequence whose first few terms are { 7 , 1 , 8 , 2 , 8 , 1 , 8 , 2 , 8 , 4 , 5 , ... } This sequence does not have an equation for the n th term. A. Alaca MATH 1005 Winter 2010 3 Let us consider the sequence { a n } = { n 2 n + 1 } . a 1 = 1 3 ≈ . 33 a 2 = 2 5 = 0 . 4 a 3 = 3 7 ≈ . 42 a 4 = 4 9 ≈ . 44 a 5 = 5 11 ≈ . 45 a 1 ≤ a 2 ≤ a 3 ≤ a 4 ≤ a 5 On the other hand, < n 2 n + 1 < n 2 n = 1 2 , for all n ≥ 1 ....

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