Homework SourseDefine a real sequence xn by x1 0 x2 1 and xn2 13 xn1 23
Define a real sequence (x_n) by x_1 = 0, x_2 = 1 and x_n+2 = 1/3 x_n+1 + 2/3 x_n for n greater than or equal to 1.Show that (x_n) is Cauchy (hence convergent). For your own amusement, see if you can determine the limit.
Solution
We have |xn+1 –xn| = | (1/3)xn + (2/3)xn-1- xn] | = |(2/3)xn-1 - (2/3)xn | = 2/3|xn-1 -xn |. This implies that |xn+1 – xn| = 2/3 |xn – xn-1|. Hence (xn) satisfies the contractive condition and therefore it satisfies the Cauchy criterion.
