Chapter 3 - CBU

3.5. THE CAUCHY CRITERION 69Example. x 1 = 1, x 2 = 2, x n = 1 2 (x n 2 + x n 1 ) for n 3.⇣(x n ) = 1, 2, 3 2 , 7 4 , 138 , 27 ⌘16 , . . . .(a) (x n ) is contractive. Thus (x n ) converges.Proof.|x n+2 x n+1 | = 1 2 (x n + x n+1 ) x n+1 = 1 2 x n(b) Note that|x n+1 x n | = 12 n 1|x 2 x 1 | = 12 n 1.(c) [To find lim(x 2n+1 ) = lim(x n ).]Thus1 + 1 2x 2n+1 x 2n 1 = 1 2 (x 2n 1 + x 2n ) x 2n 1 = 1 2 x 2n12 |x 2n x 2n 1 | = 1 2 ·12 = 12n 2 212 x n+1 = 1 2 |x n+1 x n |.2n 1.12 x 2n 1 =x 2n+1 = x 2n 1 + 12 = x 2n 1 2n 3 + 12 + 12n 3 2 = · · · =h2n 11 + 1 2 + 1 2 + 1 3 2 + · · · + 15 2 = 1 + 1 2n 1 2h1 + 1 2 + 1 2 2 + · · · + 141 + 1 2 · 1 142 2n 2 i= 1 + 1 21141 + 4 14 n 16Thus lim(x n ) = lim(x 2n+1 ) = 5 3 .n1 + 1 2 + 1 2 2 + · · · + 1 i=4 2h2n 2 1 + 1 ⇣ 1⌘ 2 ⇣ 1 n 1 i4 + + · · · +44⌘= 1 + 1 2 · 4 14 n 14 1 =! 1 + 2 3 = 5 3as n ! 1.⇤=