Introduction to Mathematical Analysis I - Second Edition

37 (b) lim n → ∞ 3 n n ! . (c) lim n → ∞ n ! n n . (d) lim n → ∞ n 2 3 n . ( Hint: see Exercise 1.3.4( c) ) . 2.1.7 Prove that if lim n → ∞ a n = ` > 0, then there exists N ∈ N such that a n > 0 for all n ≥ N . 2.1.8 I Prove that if lim n → ∞ a n = ` 6 = 0, then lim n → ∞ a n + 1 a n = 1. Is the conclusion still true if ` = 0? 2.1.9 Let { a n } be a sequence of real numbers such that lim n → ∞ a n = 3. Use Definition 2.1.1 to prove the following (a) lim n → ∞ 3 a n − 7 = 2; (b) lim n → ∞ a n + 1 a n = 4 3 ; ( Hint: prove first that there is N such that a n > 1 for n ≥ N .) 2.1.10 Let a n ≥ 0 for all n ∈ N . Prove that if lim n → ∞ a n = ` , then lim n → ∞ √ a n = √ ` . 2.1.11 Prove that the sequence { a n } with a n = sin ( n π / 2 ) is divergent. 2.1.12 I Consider a sequence { a n } . (a) Prove that lim n → ∞ a n = ` if and only if lim k → ∞ a 2 k = ` and lim k → ∞ a 2 k + 1 = ` . (b) Prove that lim n → ∞ a n = ` if and only if lim k → ∞ a 3 k = ` , lim k → ∞ a 3 k + 1 = ` , and lim k → ∞ a 3 k + 2 = ` . 2.1.13 Given a sequence { a n } , define a new sequence { b n } by b n = a 1 + a 2 + . . . + a n n . (a) Prove that if lim n → ∞ a n = ` , then lim n → ∞ b n = ` . (b) Find a counterexample to show that the converse does not hold in general. 2.2 LIMIT THEOREMS We now prove several theorems that facilitate the computation of limits of some sequences in terms of those of other simpler sequences. Theorem 2.2.1 Let { a n } and { b n } be sequences of real numbers and let k be a real number. Suppose { a n } converges to a and { b n } converges to b . Then the sequences { a n + b n } , { ka n } , and { a n b n } converge and (a) lim n → ∞ ( a n + b n ) = a + b ; (b) lim n → ∞ ( ka n ) = ka ; (c) lim n → ∞ ( a n b n ) = ab ; (d) If in addition b 6 = 0 and b n 6 = 0 for n ∈ N , then a n b n converges and lim n → ∞ a n b n = a b .

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