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
icon
Related questions
Question
100%

Please solve the screenshot in typefont (preferred) and explain each step. Thank you!

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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 6 images

Blurred answer
Knowledge Booster
Point Estimation, Limit Theorems, Approximations, and Bounds
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,