Consider the function
f(x) = x
3 and use the Newton method to analytically find expressions for the
sequence members xn when you start from the initial condition x0 = 2. Show
that the Newton method converges towards the correct solution. How often do
you have to iterate to get within 0.01 of the correct solution? @ganeshie8

Question

Consider the function
f(x) = x
3 and use the Newton method to analytically find expressions for the
sequence members xn when you start from the initial condition x0 = 2. Show
that the Newton method converges towards the correct solution. How often do
you have to iterate to get within 0.01 of the correct solution? @ganeshie8

anonymous · Answer

So far I have gotten that f'(x) = 3x^2 so 
Xn+1 = Xn - Xn^3/ 3*Xn^2 which is the same as 
Xn+1 = Xn - Xn/3 
So when X0 = 2 
X1 = 2 - 2/3  = 4/3 
X2 = 4/3 - (4/3)/3 = 8/9
X3 = 8/9 - (8/9) /3 = 16/27  

My Question is how do I show that it converges towards the correct solution and how often do you have to iterate to get within 0.01 of the correct solution?