THEOREM 6 Suppose that s n and t n are sequences such that s n t n for all n 1

Theorem 6 suppose that s n and t n are sequences such

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THEOREM 6. Suppose that { s n } and { t n } are sequences such that s n t n for all n . 1. If s n + , then t n + . 2. If t n → -∞ , then s n → -∞ . THEOREM 7. Let { s n } be a sequence of positive numbers. Then s n + if and only if 1 /s n 0 . Proof: Suppose s n → ∞ . Let > 0 and set M = 1 / . Then there exists a positive integer N such that s n > M for all n > N . Since s n > 0, 1 /s n < 1 /M = for all n > N which implies 1 /s n 0. Now suppose that 1 /s n 0. Choose any positive number M and let = 1 /M . Then there exists a positive integer N such that 0 < 1 s n < = 1 M for all n > N. that is, 1 s n < 1 M . Since s n > 0 for all n , 1 /s n < 1 /M for all n > N implies s n > M for all n > N . Therefore, s n → ∞ . Exercises 2.2 1. Prove or give a counterexample. (a) If s n s and s n > 0 for all n , then s > 0. (b) If { s n } and { t n } are divergent sequences, then { s n + t n } is divergent. (c) If { s n } and { t n } are divergent sequences, then { s n t n } is divergent. (d) If { s n } and { s n + t n } are convergent sequences, then { t n } is convergent. (e) If { s n } and { s n t n } are convergent sequences, then { t n } is convergent. (f) If { s n } is not bounded above, then { s n } diverges to + . 2. Determine the convergence or divergence of{sn}. Find any limits that exist.17
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(a) s n = 3 - 2 n 1 + n (b) s n = ( - 1) n n + 2 (c) s n = ( - 1) n n 2 n - 1 (d) s n = 2 3 n 3 2 n (e) s n = n 2 - 2 n + 1 (f) s n = 1 + n + n 2 1 + 3 n 3. Prove the following: (a) lim n →∞ n 2 + 1 - n = 0. (b) lim n →∞ n 2 + n - n = 1 2 . 4. Prove Theorem 4. 5. Prove Theorem 6. 6. Let { s n } , { t n } , and { u n } be sequnces such that s n t n u n for all n . Prove that if s n L and u n L , then t n L . II.3. MONOTONE SEQUENCES AND CAUCHY SEQUENCES Monotone Sequences Definition 4. A sequence { s n } is increasing if s n s n +1 for all n ; { s n } is decreasing if s n s n +1 for all n . A sequence is monotone if it is increasing or if it is decreasing. Examples (a) 1 , 1 2 , 1 3 , 1 4 , . . ., 1 n , . . . is a decreasing sequence. (b) 2 , 4 , 8 , 16 , . . ., 2 n , . . . is an increasing sequence. (c) 1 , 1 , 3 , 3 , 5 , 5 , . . ., 2 n - 1 , 2 n - 1 , . . . is an increasing sequence. (d) 1 , 1 2 , 3 , 1 4 , 5 , . . . is not monotonic. Some methods for showing monotonicity: (a) To show that a sequence is increasing, show that s n +1 s n 1 for all n . For decreasing, show s n +1 s n 1 for all n . The sequence s n = n n + 1 is increasing: Since s n +1 s n = ( n + 1) / ( n + 2) n/ ( n + 1) = n + 1 n + 2 · n + 1 n = n 2 + 2 n + 1 n 2 + 2 n > 1 18
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(b) By induction. For example, let { s n } be the sequence defined recursively by s n +1 = 1 + s n , s 1 = 1 . We show that { s n } is increasing. Let S be the set of positive integers for which s k +1 s k . Since s 2 = 1 + 1 = 2 > 1 , 1 S . Assume that k S ; that is, that s k +1 s k . Consider s k +2 : s k +2 = 1 + s k +1 1 + s k = s k +1 . Therefore, s k +1 S and { s n } is increasing. THEOREM 8. A monotone sequence is convergent if and only if it is bounded. Proof: Let { s n } be a monotone sequence. If { s n } is convergent, then it is bounded (Theorem 2). Now suppose that { s n } is a bounded, monotone sequence. In particular, suppose { s n } is increasing. Let u = sup { s n } and let be a positive number. Then there exists a positive integer N such that u - < s N u . Since { s n } is increasing, u - < s n u for all n > N . Therefore, | u - s n | < for all n > N and s n u .
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