HW3 Solutions

# HW3 Solutions - Math 131A Analysis Summer Session A...

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Unformatted text preview: Math 131A Analysis Summer Session A Homework 3 Solutions 1. Let ( a n ) be a bounded sequence and let ( b n ) be a sequence such that lim n →∞ b n = 0. Prove that lim n →∞ a n b n = 0. Proof. Since ( a n ) is bounded, there exists M > 0 such that | a n | < M for all n ∈ N . For all n ∈ N , we have | a n b n | = | a n || b n | < M | b n | , or equivalently,- M | b n | < a n b n < M | b n | . Since b n → 0, we have | b n | → 0 by the limit theorems. Thus ± M | b n | → 0 by the limit theorems. By the Squeeze Theorem, a n b n → 0. 2. Let p > 0 be a positive real number. Then prove lim n →∞ a n n p = if | a | ≤ 1 + ∞ if a > 1 does not exist if a <- 1 Proof. If | a | ≤ 1, the sequence a n is bounded. Indeed, | a n | = | a | n ≤ 1 for all n ∈ N . Also 1 n p → 0 for all p > 0. Thus by Problem 1, lim n →∞ a n n p = 0. If a > 1, then by the limit theorems, we have a n +1 ( n +1) p a n n p = a n +1 a n n p ( n + 1) p = a n n + 1 p = a 1 1 + 1 n p → a > 1 Thus by Homework 2, problem 7b, we have lim n →∞ a n n p = lim n →∞ a n n p = ∞ . If a <- 1, then to show the limit does not exist, it suffices to prove lim n →∞ a 2 n (2 n ) p = ∞ and lim n →∞ a 2 n- 1 (2 n- 1) p =-∞ since that implies liminf a n n p =-∞ < ∞ = limsup a n n p Since a 2( n +1)- 1 (2( n +1)- 1) p a 2 n- 1 (2 n- 1) p = a 2 n +1 (2 n +1) p a 2 n- 1 (2 n- 1) p = a 2 2 n- 1 2 n + 1 p → a 2 > 1 1 by Homework 2, problem 7b we have lim n →∞ a 2 n- 1 (2 n- 1) p = lim n →∞- a 2 n- 1 (2 n- 1) p = + ∞ Thus lim n →∞ a 2 n- 1 (2 n- 1) p =-∞ Similarly, a 2( n +1) (2( n +1)) p a 2 n (2 n ) p = a 2 n +2 (2 n +2) p a 2 n (2 n ) p = a 2 2 n + 2 2 n p → a 2 > 1 by Homework 2, problem 7b we have lim n →∞ a 2 n (2 n ) p = lim n →∞ a 2 n (2 n ) p = + ∞ 3. If lim n →∞ s n = + ∞ and k < 0, prove lim n →∞ ks n =-∞ . Proof. Define a sequence k n = k for all n ∈ N . Clearly, k n → k < 0. Thus- k n → - k > 0. By Theorem 9 . 9, lim n →∞- s n k n = lim n →∞ s n (- k n ) = + ∞ , which implies lim n →∞ ks n = lim n →∞- (- s n k n ) =-∞ . 4. If lim n →∞ s n = + ∞ and ( t n ) is a bounded sequence, prove lim n →∞ s n + t n = + ∞ . Proof. Since ( t n ) is bounded, there exists C > 0 such that- C < t n < C for all n ∈ N . Let M > 0. Since lim n →∞ s n = + ∞ , there exists N ∈ N so that s n > M + C for all n > N . Thus s n + t n > ( M + C )- C = M for all n > N ....
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HW3 Solutions - Math 131A Analysis Summer Session A...

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