In Section 4.9 we considered Newton’s method for approximating a root r of the equation f(x) = 0, and from an initial approximation x1 we obtained successive approximations x2, x3, . . . , where xn+1 = xn – f(xn)/ f’(xn)
Use Taylor’s Inequality with n = 1, a = xn, and x = r to show that if f”(x) exists on an interval l containing r, xn, and xn+1, and | f”(x) | < M. | f’(x) | > K, for all x ε l, then | xn+1 – r | < M/2K | xn – r |2
[This means that if xn is accurate to decimal places, then xn+1 is accurate to about 2d decimal places. More precisely, if the error at stage is at most 10–m, then the error at stage n + 1 is at most (M/2K) 10–2m.]