The purpose of this exercise is to establish the convergence of an iterative process for approximating square roots. Define a sequence recursively as follows: X1 = 2 1 Xn z(Xn-1 2 + Xn-1 A) Show that xn > v2 B) Show that the sequence is decreasing C) Show that the sequence converges to v2

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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The purpose of this exercise is to establish the convergence of an iterative process for approximating
square roots. Define a sequence recursively as follows:
X1 = 2
1
2
(xn-1 +
2
Xn
Xn-1
A) Show that xn 2 v2
B) Show that the sequence is decreasing
C) Show that the sequence converges to V2
D) Let ɛ
X, - V2 and show that ɛn+1
which would mean letting o = 2v2 which
2xn
makes ɛn+1 < o()2n
(n = 1,2,3, ..)
E) Generalize this process to find a sequence that converges to Va where x is any positive real
number
F) Generalize part (D) WITHOUT ANY PROOF and use it with x1 = 2 and a = 3 to determine
E1 , ... , E6
Note: this will show how rapid the convergence of the recursion formula is in calculating
square roots
Transcribed Image Text:The purpose of this exercise is to establish the convergence of an iterative process for approximating square roots. Define a sequence recursively as follows: X1 = 2 1 2 (xn-1 + 2 Xn Xn-1 A) Show that xn 2 v2 B) Show that the sequence is decreasing C) Show that the sequence converges to V2 D) Let ɛ X, - V2 and show that ɛn+1 which would mean letting o = 2v2 which 2xn makes ɛn+1 < o()2n (n = 1,2,3, ..) E) Generalize this process to find a sequence that converges to Va where x is any positive real number F) Generalize part (D) WITHOUT ANY PROOF and use it with x1 = 2 and a = 3 to determine E1 , ... , E6 Note: this will show how rapid the convergence of the recursion formula is in calculating square roots
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