5: Convergence of Newton's method for ()= 3 (20 pts.) We want to study the order of convergence of Newton's method when ( )= ( )=0 at the root. Step 1. Write down one iteration of the Newton's method from to 1 . Use the standard error analysis approach where = and similarly for 1 . Insert in the iteration formula and use Taylor series expansion to derive the equation for the errors at each iteration with at least two terms. You can consult your lecture notes for this part. Step 2. Now for the function ()= 3 substitute the derivatives and simplify the error equation from Step 1. Step 3. The roots of are at =0 . Take the limit of the equation from Step 2 as 0 and simplify. What is largest remaining term in the error? From that, identify what is the order of convergence of Newton's method for this example