Sequences - PART II. SEQUENCES OF REAL NUMBERS II.1....

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PART II. SEQUENCES OF REAL NUMBERS II.1. CONVERGENCE Defnition 1. A sequence is a real-valued function f whose domain is the set positive integers ( N ). The numbers f (1) ,f (2) , ··· are called the terms of the sequence. Notation Function notation vs subscript notation: f (1) s 1 (2) s 2 , ( n ) s n , . In discussing sequences the subscript notation is much more common than functional notation. We’ll use subscript notation throughout our treatment of analysis. SpeciFying a sequence There are several ways to specify a sequence. 1. By giving the function. For example: (a) s n = 1 n or { s n } = ± 1 n ² . This is the sequence { 1 , 1 2 , 1 3 , 1 4 ,..., 1 n ,... } . (b) s n = n - 1 n . This is the sequence { 0 , 1 2 , 2 3 , 3 4 n - 1 n } . (c) s n =( - 1) n n 2 . This is the sequence {- 1 , 4 , - 9 , 16 ( - 1) n n 2 } . 2. By giving the ±rst few terms to establish a pattern, leaving it to you to ±nd the function. This is risky – it might not be easy to recognize the pattern and/or you can be misled. (a) { s n } = { 0 , 1 , 0 , 1 , 0 , 1 } . The pattern here is obvious; can you devise the function? It’s s n = 1 - ( - 1) n ) 2 or s n = ³ 0 ,n odd 1 even (b) { s n } = ± 2 , 5 2 , 10 3 , 17 4 , 26 5 ² ,s n = n 2 +1 n . (c) { s n } = { 2 , 4 , 8 , 16 , 32 } . What is s 6 ? What is the function? While you might say 64 and s n =2 n , the function I have in mind gives s 6 = π/ 6: s n n +( n - 1)( n - 2)( n - 3)( n - 4)( n - 5) ´ π 720 - 64 120 µ 3. By a recursion formula. For example: (a) s n +1 = 1 n s n 1 = 1. The ±rst 5 terms are ± 1 , 1 2 , 1 6 , 1 24 , 1 120 ² . Assuming that the pattern continues s n = 1 n ! . (b) s n +1 = 1 2 ( s n +1) 1 = 1. The ±rst 5 terms are { 1 , 1 , 1 , 1 , 1 } . Assuming that the pattern continues s n = 1 for all n ; { s n } is a “constant” sequence. 13
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Defnition 2. A sequence { s n } converges to the number s if to each ±> 0 there corresponds a positive integer N such that | s n - s | for all n>N. The number s is called the limit of the sequence. Notation { s n } converges to s ” is denoted by lim n →∞ s n = s, or by lim s n = s, or by s n s. A sequence that does not converge is said to diverge . Examples Which of the sequences given above converge and which diverge; give the limits of the convergent sequences. THEOREM 1. If s n s and s n t , then s = t . That is, the limit of a convergent sequence is unique. ProoF: Suppose s ± = t . Assume t>s and let ± = t - s . Since s n s , there exists a positive integer N 1 such that | s - s n | <±/ 2 for all n>N 1 . Since s n t , there exists a positive integer N 2 such that | t - s n | 2 for all 2 . Let N = max { N 1 ,N 2 } and choose a positive integer k>N . Then t - s = | t - s | = | t - s k + s k - s |≤| t - s k | + | s - s k | < ± 2 + ± 2 = ± = t - s, a contradiction. Therefore, s = t . THEOREM 2. If { s n } converges, then { s n } is bounded. ProoF: Suppose s n s . There exists a positive integer N such that | s - s n | < 1 for all . Therefore, it follows that | s n | = | s n - s + s s n - s | + | s | < 1+ | s | for all Let M = max {| s 1 | , | s 2 | , ..., | s N | , | s |} . Then | s n | <M for all n . Therefore { s n } is bounded. THEOREM 3. Let { s n } and { a n } be sequences and suppose that there is a positive number k and a positive integer N such that | s n |≤ ka n for all If a n 0 , then s n 0 .
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Sequences - PART II. SEQUENCES OF REAL NUMBERS II.1....

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