Chapter 2.5, Problem 2ES

What is the Taylor series for f(x)=cos(2x) at x=pi/4 and its interval of convergence?

Use the equation 1/(1-x) =Σ xn (from n=0 to infinity) for the absolute value of x is less than 1 to expand the function 4/1-x in a power series with the center at c=-2  Then determine the interval of convergence

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?