3.4. Consider the problem minimize f(x) = x' - 1. Solve this problem using Newton's method. Start from xo = 4 and perform three iterations. Prove that the iterates converge to the solution. What is the rate of convergence? Can you explain this?

7. Let f(x) = E (-1)"x2n+1 Zn=0 Find the interval of convergence of f. (2n+1)!

Solve Part (iv), (v) and (vi) only

3. Define fn on (0, 1) by fn(x) = x". Show that fn converges to the constant function 0, but that this convergence is not uniform.

19. Prove that if r is a zero of multiplicity k of the function f, then quadratic convergence in Newton's iteration will be restored by making this modification: Xn+1 = Xn-kf(xn)/f'(xn)

Help me fast so that I will give Upvote.

For problem 1: a) You need to let f(x) = 1/x(lnx)^2 and show that it is positive, continuous, and decreasing just like we did in class -- see your notes. b) When computing an improper integral, you need to rewrite the integral in terms of limits  -- see your notes from section 8.8

. Suppose f(x) = en(x – a)" is differentiable on the interval (a – 4, a + 4) and has the same interval n=0 of convergence. Answer the following questions (no justification necessary). (a) What is the radius of convergence for f'(x)? Radius : (b) What is the radius of convergence for | f(x) dx? Radius :

Suppose to solve for a root of f(x), i.e., f(x*) = 0, we use the iteration Xk+1 = ¢(xk) where o(x) and f(x) given functions with as many continuous derivatives as you require. Show that if |O'(x*)| < 1 then for |xo – x*| sufficiently small the iteration xk+1 = produces a sequence that converges to x*. Note that the iteration ø(x) is called a contraction mapping in the neighborhood of x* when |ø'(x*)| < 1. -