Newton-Raphson Method and Prove

Question

I have studying numerical analysis and came across this question.

Find the approximate value of $\sqrt5$ by using Newton-Raphson method. Take initial approximation as $2$ and $\epsilon = 10^{-6}$. You also prove that Newton-Raphson iteration scheme is second order convergent technique.

Can anybody help me ?

Answer 1

Hint Let $x=\sqrt5$ then $f(x)=x^2-5=0$ The Newton–Raphson method for this equation is given by $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$.