Newton’s method for k-Frechet differentiable operators

We determine a solution x* of the equation F ( x ) = 0 , $$ F(x) = 0, $$ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315153469/276b2768-c97f-4f2f-95fc-28fbbbbbe91f/content/math2_1.tif"/>