Chapter 3 - CBU

3.3. MONOTONE SEQUENCES 53Example. Determine whether lim(x n ) exists and, if so, its value wherex 1 = 1 and x n+1 = p 1 + x n for n 1.Solution.x 2 = p 1 + 1 = p 2, x 3 =(a) [Show monotone increasing.]q1 + p 2, x 4 =x 1 < x 2 since 1 < p 2. Assume x n apple x n+1 .Then x n+1 = p 1 + x n apple p 1 + x n+1 = x n+2 ,so by induction x n apple x n+1 8 n 2 N.Thus (x n ) is increasing.(b) [Show (x n ) is bounded above by 2 using induction.]x 1 = 1 < 2. Suppose x n apple 2. Thenr1 +x n+1 = p 1 + x n apple p 1 + 2 = p 3 < p 4 = 2.Thus, by induction, x n apple 2 8 n 2 N,and so 2 is an upper bound of (x n ).(c) Thus lim(x n ) = x for some x 2 R by the MCT.Since (x n+1 ) is a tail of (x n ), lim(x n+1 ) = x also. Thenq1 + p 2 , . . .x = lim(x n+1 ) = lim( p 1 + x n ) = p lim(1 + x n ) =plim(1) + lim(xn ) = p 1 + x =)Since 1 p52x 2 = 1 + x =) x 2 x 1 = 0 =) x = 1 ± p 5.2< 0, we conclude x = lim(x n ) = 1 + p 5. ⇤2