If x is the estimate of the fixed point at the nth iteration, then using Taylor expansion of the function g(x) about x* determine the convergence rate. Does the result of this analysis agree with that in 1.b? [8] 23 3! (You may need: sin(x) = x − 3¾³ + O(x³) and cos(x) = 1 − 22² + O(x4).) - -

Suppose that ∞ n = 0 anxn converges to a function y such that y'' − 2y' + y = 0 where y(0) = 0 and y'(0) = 1. Find a formula that relates an + 2, an + 1, and

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

Find the lower bound for the radius of convergence for(x2 + 6x+ 10)y′′+ (x2 −4)y′+ x2y= 0 about x0 = 0

Suppose f(x) > 0 and g(x) > 0 and that | f(æ)dz diverges. If f(x) > g(x), what can we conclude about | g(z)dæ? It converges It diverges O We can make no conclusions

Write down the Taylor's series expansion of f(x) = cos x at x = in 3 terms of f(x), and its derivatives at x = . Compute the approximations 4 from the zeroth order to the fifth order and also state the absolute error in each case.

Consider the function, f(x) = x^3 − x^2 − 9x + 9. Answer the following: (a)  State the exact roots of f(x).(b)  Construct three different fixed point functions g(x) such that f(x) = 0. (Make sure that one of theg(x)’s that you constructed converges to at least a root).(c) Find the convergence rate/ratio for g(x) constructed in previous part and also find which root it isconverging to?(d)  Find the approximate root, x⋆,  of the above function using fixed point iterations up to 4 significantfigures within the error bound of 1 × 10^−3 using X0 = 0 and any fixed point function g(x) from part(b) thatconverges to the root(s).