Chapter 5.1, Problem 54E

Newton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0 and successively defines a sequence $z_{n+1}=x_n-\frac{f\left(x_n\right)}{f\text{'}\left(x_n\right)}$ . For the given choice of f and x₀. write out the formula for $x_{n+1}$ . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x_nagrees with x up to four decimal places.

54. $\left[\text{T}\right]f\left(x\right)=x^2-2,x_0=1$