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
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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
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(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
MATH fa hmw solutions
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
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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)
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(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
Practice Solution
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
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17 nov 2008 · Exercise 1 (#13) Let x1 = 2 and xn+1 =2+1/xn Then xn is contractive, and lim xn =1+
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(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
Problem
Let x1 and x2 have the joint p.d.f f(x1x2) = x1 +x2 with 0 ? x1
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3) Let x1 ? 2 and xn+1 := 1 +. ? xn ? 1n ? N. Show that (xn) is decreasing and bounded below by 2. Find the limit. Solution We are given x1 ? 2.
Nov 8 2021 Let X1 and X2 be independent normal random variables
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Question: Let X1 and X2 have independent distributions b(n1 p) and b(n2
Let X1 and X2 be independent with normal distributions N(61) and N(7
Let X1 and X2 be independent random variables with probability density functions f1(x1)=2x1 0< x1< 1
Var(x1) = 54; Var(X2) = 110. Let Y = 5X1 - 6X2. What is the variance of Y? This problem has been solved! See
1 3 3 3) Let x1 ? 2 and xn+1 := 1 + ? xn ? 1n ? N Show that (xn) is decreasing and bounded below by 2 Find the limit Solution We are given x1
2 Let xn = (?1)n for all n ? N Show that the sequence (xn) does not (b) x1 = ? 2 and xn+1 = ? 2xn for n ? N (c) x1 = 1 and xn+1 = 4+3xn
Question: Let x1 = 1 and xn+1 =3xn2 for n > or = 1 a Show that if a = lim xn then a = 1/3 or a= 0b Does lim xn exist? Explain c
10 nov 2008 · Exercise 1 (# 4) Let x1 = 1 and xn+1 = ? 2 + xn Then lim xn = 2 This equation has two solutions namely L = 2 L = ?1 Since xn > 0
Ex 2(g) Let x1 = 1 and xn+1 = ( n n+1 )x2 n for all n ? N Examine whether the sequence (xn) is convergent Also find the limit if it is convergent
3xn+1 ? 2xn = 8 ? 3xn+1 = 2xn + 8 ? xn+1 = 2 3 xn + x1 = x0x2 = 2x1 + 1x3 = 3x2 + 2x4 = 4x3 + 3??? in the form xn+1 Let S0 = 0 and Sn+1
xn = 9 · 3n - 5 · 4n Page 9 Example 9–6: Find the solution of the difference equation: xn+2 - 12xn+1 + 36xn = 0 x0 = 5 x1 = 18 Solution: Let's insert xn =
Every sequence in the closed interval [ab] has a subsequence in R that converges to some point in R Proof Assume a ? xn ? b for n = 12??? By Theorem 1 4
Given a > 0 define a sequence {xn} of real numbers inductively by setting x1 = 1 a and xn+1 = 1 a + xn i e xn = 1 a + 1 a + 1 a + (a) Is {xn}
SEQUENCES (1) Let (xn) be a sequence such that xn ? x and xn ? y Prove that x = y (2) Investigate the Convergence of the Sequence (xn) where (a) xn =
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EXERCISES IN STATISTICS Series A No. 6 1. Let x1 and x2 have