Sequences - Copy (2)

Sequences - Copy (2) - A Alaca MATH 1005 Winter 2010 2...

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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 = { 0 , 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.
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A. Alaca MATH 1005 Winter 2010 3 Let us consider the sequence { a n } = { n 2 n + 1 } . a 1 = 1 3 0 . 33 a 2 = 2 5 = 0 . 4 a 3 = 3 7 0 . 42 a 4 = 4 9 0 . 44 a 5 = 5 11 0 . 45 a 1 a 2 a 3 a 4 a 5 On the other hand, 0 < n 2 n + 1 < n 2 n = 1 2 , for all n 1 . This tells us that, as n becomes large the terms of the sequence are getting larger, but they can not exceed 1 / 2. With other words, as n becomes large, the terms of the sequence { n 2 n + 1 } are approaching 1 / 2. We indicate this fact by writing lim n →∞ n 2 n + 1 = 1 2 The notation lim n →∞ a n = L means the terms of the sequence { a n } can be made arbi- trarily close to L by taking n sufficiently large.
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