Introduction to Mathematical Analysis I - Second Edition

46 2.3 MONOTONE SEQUENCES Exercises 2.3.1 I Let a 1 = √ 2. Define a n + 1 = p a n + 2 for n ≥ 1 . (a) Prove that a n < 2 for all n ∈ N . (b) Prove that { a n } is an increasing sequence. (c) Prove that lim n → ∞ a n = 2. 2.3.2 B Prove that each of the following sequences is convergent and find its limit. (a) a 1 = 1 and a n + 1 = a n + 3 2 for n ≥ 1. (b) a 1 = √ 6 and a n + 1 = √ a n + 6 for n ≥ 1. (c) a n + 1 = 1 3 2 a n + 1 a 2 n , n ≥ 1 , a 1 > 0. (d) a n + 1 = 1 2 a n + b a n , b > 0. 2.3.3 B Prove that each of the following sequences is convergent and find its limit. (a) √ 2; p 2 √ 2; q 2 p 2 √ 2; · · · (b) 1 / 2; 1 2 + 1 / 2 ; 1 2 + 1 2 + 1 / 2 ; · · · 2.3.4 Prove that the following sequence is convergent: a n = 1 + 1 2! + 1 3! + · · · + 1 n ! , n ∈ N . 2.3.5 B Let a and b be two positive real numbers with a < b . Define a 1 = a , b 1 = b , and a n + 1 = p a n b n and b n + 1 = a n + b n 2 for n ≥ 1 . Show that { a n } and { b n } are convergent to the same limit. 2.3.6 Prove the following using Definition 2.3.2 . (a) lim n → ∞ 2 n 2 + n + 1 n − 2 = ∞ . (b) lim n → ∞ 1 − 3 n 2 n + 2 = − ∞ . 2.3.7 Prove parts (b) , (c) , and (d) of Theorem 2.3.6 . 2.3.8 Prove Theorem 2.3.7 .