50 2.3 Monotone Sequences (c) Prove that limn →∞ an = 2. 2.3.2 ▷ Prove that each of the following sequences is convergent and fnd its limit. an + 3 (a) a1 = 1 and an+1 = for n ≥ 1. 2 √ √ (b) a1 = 6 and an+1 = an + 6 for n ≥ 1. 1 (c) a1 > 1 and an+1 = 2 − for n ≥ 1. an 1 1 (d) a1 > 0 and an+1 = 2an + for n ≥ 1. a2 3 n √ 1 b (e) a1 > b for b > 0 and an+1 = an + for n ≥ 1. 2 an 2.3.3 ▷ Prove that each of the following sequences is convergent and fnd its limit. q √ p √ p √ (a) 2; 22; 2 22; ··· 1 1 (b) 1/2; ; ;··· 2 + 1/2 2 + 1 2+ 1/2 2.3.4 Prove that for every real number x there is a strictly monotone sequence of rational numbers {rn} which converges to x. (Hint: use the density property of Q, Theorem 1.6.3) 2.3.5 Prove that for every real number x there is a strictly monotone sequence of irrational numbers {sn} which converges to x. (Hint: use the density of the irrational numbers, Theorem 1.6.5.) 2.3.6 Prove that the following sequence is convergent: 1 1 1 an = 1 + + + ··· + , n ∈ N. 2! 3! n! 2.3.7 ▷ Let a and b be two positive real numbers with a < b. Defne a1 = a, b1 = b, and p an + bn an+1 = anbn and bn+1 = for n ≥ 1. 2 Show that {an} and {bn} converge to the same limit. 2.3.8 Prove the following using Defnition 2.3.2. 2n2 + n+ 1 (a) lim = ∞. n→∞ n− 2 2 1− 3n (b) lim = −∞. n→∞ n + 2 2.3.9 Prove parts (ii), (iii), and (iv) of Theorem 2.3.4. 2.3.10 Prove Theorem 2.3.5.
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