Binomial expansion of Newton's method

We extend the Newton's method and show the extended Newton's method leads to the binomial expansion of Newton's method that the convergences become the quadratic and linearly. In case of the quadratic convergence, we give the convergence comparison of the binomial expansion of Newton's method and Newton's method. And we give convergence comparisons of the binomial expansion of Newton's method and Newton's method using the curvature and convex-concave of curve. Next we give examples of numerical calculations of the binomial expansion of Newton's method.