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Consider the sequence In+1 n = 0, 1, 2,... (a) Show that this sequence is generated by Newton's method used to estimate √K. eu-e-u (b) Make the substitution n = √K coth (bn), where coth u = e te-u is the hyperbolic cotangent, to convert the original sequence to bn+1 = 2bn. Solve this sequence for bn (i.e. for some initial term bo, find b, as an explicit function of n). Use your solution to prove that the original sequence {n} converges to VK, In In Taylor polynomials lim n = √K 818 (c) Find a fraction that approximates √5. How did you choose the initial term in the sequence zo? Use a calculator to estimate the error in your approximation.

Consider the sequence In+1 n = 0, 1, 2,... (a) Show that this sequence is generated by Newton's method used to estimate √K. eu-e-u (b) Make the substitution n = √K coth (bn), where coth u = e te-u is the hyperbolic cotangent, to convert the original sequence to bn+1 = 2bn. Solve this sequence for bn (i.e. for some initial term bo, find b, as an explicit function of n). Use your solution to prove that the original sequence {n} converges to VK, In In Taylor polynomials lim n = √K 818 (c) Find a fraction that approximates √5. How did you choose the initial term in the sequence zo? Use a calculator to estimate the error in your approximation.

Advanced Engineering Mathematics
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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1. Consider the sequence K In Intl n = 0, 1, 2,... (a) Show that this sequence is generated by Newton's method used to estimate VK. (b) Make the substitution In = √K coth (bn), where coth u = is the hyperbolic cotangent, to convert the original sequence to bn+1 = 2bn. Solve this sequence for bn (i.e. for some initial term bo, find bn, as an explicit function of n). Use your solution to prove that the original sequence {n} converges to VK, e" +e-u eu-e-u lim In = n4x = √K Taylor polynomials (c) Find a fraction that approximates √5. How did you choose the initial term in the sequence ro? Use a calculator to estimate the error in your approximation.
Transcribed Image Text:1. Consider the sequence K In Intl n = 0, 1, 2,... (a) Show that this sequence is generated by Newton's method used to estimate VK. (b) Make the substitution In = √K coth (bn), where coth u = is the hyperbolic cotangent, to convert the original sequence to bn+1 = 2bn. Solve this sequence for bn (i.e. for some initial term bo, find bn, as an explicit function of n). Use your solution to prove that the original sequence {n} converges to VK, e" +e-u eu-e-u lim In = n4x = √K Taylor polynomials (c) Find a fraction that approximates √5. How did you choose the initial term in the sequence ro? Use a calculator to estimate the error in your approximation.
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