Sequences - PART II SEQUENCES OF REAL NUMBERS II.1 CONVERGENCE Denition 1 A sequence is a real-valued function f whose domain is the set positive

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

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PART II. SEQUENCES OF REAL NUMBERS II.1. CONVERGENCE Definition 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 , f (2) s 2 , · · · , f ( 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 . 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 first few terms to establish a pattern, leaving it to you to find 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 = 0 , n odd 1 , n even (b) { s n } = 2 , 5 2 , 10 3 , 17 4 , 26 5 , . . . , 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 = 2 n + ( n - 1)( n - 2)( n - 3)( n - 4)( n - 5) π 720 - 64 120 3. By a recursion formula. For example: (a) s n +1 = 1 n + 1 s n , s 1 = 1. The first 5 terms are 1 , 1 2 , 1 6 , 1 24 , 1 120 , . . . . Assuming that the pattern continues s n = 1 n ! . (b) s n +1 = 1 2 ( s n + 1) , s 1 = 1. The first 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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Definition 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 n > N 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 n > N . Therefore, it follows that | s n | = | s n - s + s | ≤ | s n - s | + | s | < 1 + | s | for all n > N. Let M = max {| s 1 | , | s 2 | , . . ., | s N | , 1 + | 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 | ≤ k a n for all n > N.
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