Approximation of the v th Root of N

We present a class of functions g K ( w ), K ≥ 2, for which the recursive sequences w n + 1 = g K ( w n ) converge to N 1/ v with relative error . Newton's method results when K = 2. The coefficients of the g K ( w ) form a triangle, which is Pascal's for v = 2. In this case, if w 1 = x 1 / y 1 , where x 1 , y 1 is the first positive solution of Pell's equation x 2 − Ny 2 = 1, then w n + 1 = x n + 1 / y n + 1 is the K n p th or 2 K n p th convergent of the continued fraction for , its period p being even or odd.