If x is the estimate of the fixed point at the nth iteration, then using Taylorexpansion of the function g(x) about x* determine the convergence rate. Does theresult of this analysis agree with that in 1.b? [8]233!(You may need: sin(x) = x − 3¾³ + O(x³) and cos(x) = 1 − 22² + O(x4).)--

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Answered: If x is the estimate of the fixed point…

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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Find the simple continued fraction representation of (1 + sqrt6)/2 and show that it is periodic. (Do not use deciomal approximations in your proof!) Then compute the first eight convergents and find the best rational approximation with a denominator less than 30. (Justify your answer).
To find the root of the function f(x), write down the mapping function, g(x), for Newton’s method. Using x0 = π/3 as the initial guess, perform the first four iterations using Newton’s method. Also calculate the ratio |xn/xn−1| at every step. What does this ratio denote and what does its limiting value tell you about the convergence rate of Newton’s method?
Write down the 4 by 4 matrix S that scales by a constant c. Multiply ST and also TS, where T is translation by (1, 4, 3). To blow up the picture around the center point (1, 4, 3), would you use vST or vT S?
8. The Taylor series about x = 3 for a certain function f converges to f(x) for all x in the interval of convergence. The nth derivative off at x = 3 is given by (-1)" n! and ƒ (3) = -3 5" (n+3) f(n) (3)= 1 (c) Show that the third-degree Taylor polynomial approximates f (4) with an error less than 4000
12. Find the radius and interval of convergence of the power series n=0 (x-4) √5" n 18. Find the 3th -degree Taylor polynomial centered at x=8 for the function f(x)=√x. Use the polynomial to approximate 3/8.02. Compare the value to the value obtained by your calculator. 19. Find the first three non-zero terms of the Taylor series for f(x) = secx centered at a=. Please show all work. 20. How large must n be to approximate values of sin x within 0.01 on the interval (-1,1) using the series sin(x) = 8 Σ (−1)"x2n+1 ? (2n+1)! n=0
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3. Show that II (1+22) is absolute convergent if and only if |z| < 1. Show k=0 that the product converges to 4. Show that 1+sin( is absolute and uniform convergent for all 2. k=0 5. Show that Res(T(2);-n) = (-1)" n!
4. Let f be differentiable on R with a = sup{|ƒ'(x)| : x € R} < 1. f(sn-1) for n ≥ 1. Prove that (i) Pick a number so € R and inductively define sn = {n} is a convergent sequence in R. (ii) Show that f has a fixed point, that is, f(s) = s for some s E R.

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