Given the function f : (0, 0) → R where 2² +2 f(x) = 2.x for all r € (0, o0), consider a sequence r = {rn}=1 defined by x1 = 1, xn+1 = f(x,) for all n > 1. (a) Prove that if x converges, then its limit is r = /2. (Hint: What must be true at or close to the limit?) (b) First, show that, for any two real numbers a and b, a? + b² > 2ab holds. Then, by applying this formula, show that rn 2 V2 for all n 2 2. (c) Compute the value of rn+1-In as a function of xn and rn-1. Based on this computation, show that r is a decreasing function.
Given the function f : (0, 0) → R where 2² +2 f(x) = 2.x for all r € (0, o0), consider a sequence r = {rn}=1 defined by x1 = 1, xn+1 = f(x,) for all n > 1. (a) Prove that if x converges, then its limit is r = /2. (Hint: What must be true at or close to the limit?) (b) First, show that, for any two real numbers a and b, a? + b² > 2ab holds. Then, by applying this formula, show that rn 2 V2 for all n 2 2. (c) Compute the value of rn+1-In as a function of xn and rn-1. Based on this computation, show that r is a decreasing function.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Given the function f : (0, 0) → R where
2² +2
f(x) =
2.x
for all r € (0, o0),
consider a sequence r = {rn}=1 defined by
x1 = 1, xn+1 = f(x,) for all n > 1.
(a) Prove that if x converges, then its limit is r = /2. (Hint: What must be true at or close
to the limit?)
(b) First, show that, for any two real numbers a and b, a? + b² > 2ab holds. Then, by
applying this formula, show that rn 2 V2 for all n 2 2.
(c) Compute the value of rn+1-In as a function of xn and rn-1. Based on this computation,
show that r is a decreasing function.
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