50 2.3 Monotone Sequences (c) Prove that limn→∞an =2. 2.3.2 ▷Prove that each of the following sequences is convergent and find its limit. (a) a1 =1 and an+1 = an +3 2 for n≥1. (b) a1 =√6 and an+1 =√an +6 for n≥1. (c) a1 >1 and an+1 =2− 1 an for n≥1. (d) a1 >0 and an+1 = 1 3 2an + 1 a2 n for n≥1. (e) a1 >√bfor b>0 and an+1 = 1 2 an + b an for n≥1. 2.3.3 ▷Prove that each of the following sequences is convergent and find its limit. (a) √2;p2√2;q2p2√2;· · · (b) 1/2; 1 2+1/2 ; 1 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: an =1+ 1 2! + 1 3! +· · ·+ 1 n! , n∈N. 2.3.7 ▷Let aandbbe two positive real numbers with a<b. Define a1 =a, b1 =b, and an+1 =panbn andbn+1 = an +bn 2 for n≥1. Show that {an}and{bn}converge to the same limit. 2.3.8 Prove the following using Definition 2.3.2. (a) lim n→∞ 2n2 +n+1 n−2 =∞. (b) lim n→∞ 1−3n2 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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