Lesson 8.pdf - 8 Monotonic and Cauchy Sequences 1 Monotonic and Cauchy Sequences The monotonicity Definition 1(sn is increasing if sn ≤ sn 1 ∀n 2(sn

Lesson 8.pdf - 8 Monotonic and Cauchy Sequences 1 Monotonic...

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8. Monotonic and Cauchy Sequences July 17, 2019 1 Monotonic and Cauchy Sequences The monotonicity Definition 1. ( s n ) is increasing if s n s n +1 n. 2. ( s n ) is decreasing if s n s n +1 n. 3. ( s n ) is monotonic if ( s n ) is increasing or decreasing. [Examples] 1. ( s n ) = (1 , 2 , 3 , 4 , ... ) is an increasing sequence. 2. ( s n ) = (2 , 4 , 8 , 16 , ..., 2 n , ... ) is an increasing sequence. 3. ( s n ) = (1 , 2 , 3 , 3 , 5 , 5 , ..., 2 n + 1 , 2 n + 1 , ... ) is an increasing increasing. 4. ( s n ) = (1 , 1 2 , 3 , 1 4 , 5 , 1 8 , .... ) is not monotonic. How to show monotonicity —– I The first way to show monotonicity for a sequence is to show s n +1 s n 1 for all n or 1 for all n . [Examples] 70
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1. Show that the sequence ( s n ) = ( n n +1 ) is increasing To show: s n +1 s n 1 for all n . In fact s n +1 s n = n + 1 n + 2 · n + 1 n = ( n + 2) 2 n ( n + 2) = n 2 + 2 n + 1 n 2 + 2 n 1 , integer n 1 . Then ( s n ) is increasing. 2. Show that the sequence ( s n ) = ( 2 n n ! ) is decreasing To show: s n +1 s n 1 for all n . In fact s n +1 s n = 2 n +1 ( n + 1)! · n ! 2 n = 2 n + 1 1 integer n 1 . Then ( s n ) is decreasing. How to show monotonicity —– II The second way to show monotonicity for a sequence is to use mathematical induction. [Examples] 1. Let ( s n ) be defined by s n +1 = 1 + s n , s 1 = 1 . Show ( s n ) is increasing. Let S be the set of positive integers for which s k +1 s k . (a) 1 S because s 2 = 1 + 1 = 2 > 1. (b) Assume k S : s k +1 s k . (c) To show: k + 1 S , i.e., s k +2 s k . In fact, by the assumption, s k +2 = 1 + s k +1 1 + s k = s k +1 . Theorem 1.1 A monotonic sequence ( s n ) is convergent ⇐⇒ it is bounded. 71
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Proof: (= ) By Theorem 0.2 in § 6, a convergent sequence is bounded. ( =) Suppose that ( s n ) is monotonic and bounded. Assume that ( s n ) is increasing (similar proof if ( s n ) is decreasing). We want to prove that ( s n ) is convergent. Since the set { S n } bounded, let us denote α = sup { s n } . We’ll show that this α is the limit. In fact, for any > 0, by the footnote of the definition of supremum, there exists an element s N ∈ { s n } such that α - < s N α. Since ( s n ) is increasing, s N s n α for all n > N . Therefore α - < s n α < α + holds for all n > N , i.e., | s n - α | < , n > N . Then s n α . [Example] 1. Let s n +1 = 3 - 1 s n with s 1 = 1. What is the limit lim n →∞ s n ? By Theorem 1.1, if we can show that s n is increasing and bounded, then the limit of s n exists. To show s n increasing, i.e., s n s n +1 for all n 1, i.e., s n 3 - 1 s n , n 1 .
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  • Fall '08
  • Staff
  • Mathematical analysis, Natural number, Limit of a sequence, Cauchy sequence

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