1 3 3 3) Let x1 ≥ 2 and xn+1 := 1 + √ xn − 1,n ∈ N Show that (xn) is decreasing and bounded below by 2 Find the limit Solution We are given x1 ≥ 2
2 Let xn = (−1)n for all n ∈ N Show that the sequence (xn) does not converge 3 (b) x1 = √ 2 and xn+1 = √ 2xn for n ∈ N (c) x1 = 1 and xn+1 = 4+3xn
(c) By part (a), s = limsn ≥ a and by part (b) s ≤ b, so s ∈ [a, b] D Problem 2 Let x1 = 1 and xn+1 = 3x2 n for n ≥ 1 (a) Show if a = limxn, then a = 1 3 or a = 0
10 nov 2008 · 1 Section 3 3 Exercise 1 (# 4) Let x1 = 1 and xn+1 = √ 2 + xn Then lim xn = 2 First we show that xn is increasing by using an induction
1 Let x1 := 8 and xn+1 := 1 2 xn + 2 for n ∈ N Show that (xn) is bounded and monotone Find the limit Proof First, let's show that it is monotone (decreasing)
(xn) such that the sequence (yn) is convergent, where yn = xn + 1 n Ex 2(g) Let x1 = 1 and xn+1 = ( n Ex 2(h) Let a, b ∈ R, x1 = a, x2 = b and xn+2 = 1 2 −3
10 nov 2008 · 1 Section 3 3 Exercise 1 (# 4) Let x1 = 1 and xn+1 = √ 2 + xn Then lim xn = 2 First we show that xn is increasing by using an induction
17 nov 2008 · Exercise 1 (#13) Let x1 = 2 and xn+1 =2+1/xn Then xn is contractive, and lim xn =1+
(1) Let (xn) be a sequence such that xn → x and xn → y bounded and monotone, and find its limit (a) x1 = 1 and xn+1 = 4+3xn 3+2xn (b) xn+1 = 1 2 ( xn + a