Approximation of the vth Root of N

We present a class of functions gK(w), K≥ 2, for which the recursive sequences wn+ 1= gK(wn) converge to N1/vwith relative error . Newton's method results when K= 2. The coefficients of the gK(w) form a triangle, which is Pascal's for v= 2. In this case, if w1= x1/y1, where x1, y1is the first positive solution of Pell's equation x2− Ny2= 1, then wn+ 1= xn+ 1/yn+ 1is the Knpth or 2Knpth convergent of the continued fraction for , its period pbeing even or odd.