Convergence of Sequences of Approximate Solutions by Newton Method and its Elementary New Proof

As the method of finding the approximate solutions of nonlinear equation, the Newton or the secant method is well known. But, the convergence of the approximate sequence {x_n} is not necessarily assured, Indeed, in many cases, the sequence {x_n} does not converge to the true solution. The classical proof of the convergence of the approximate sequence {x_n} is done with the help of the principle of contraction mapping under some additional conditions. The aim of this paper is to give an elementary new proof of the convergence of approximate sequence without using the principle of contraction mapping. We should like to remark that our new method is applicable to the proof of the more complicated case of the secant method