A recurrence relation for the square root

Abstract

AbstractThe behavior of the sequence xn + 1 = xn(3N − xn2)2N is studied for N > 0 and varying real x0. When 0 < x0 < (3N)12 the sequence converges quadratically to N12. When x0 > (5N)12 the sequence oscillates infinitely. There is an increasing sequence βr, with β−1 = (3N)12 which converges to (5N)12 and is such that when βr < x0 < βr + 1 the sequence {xn} converges to (−1)rN12. For x0 = 0, β−1, β0,… the sequence converges to 0. For x0 = (5N)12 the sequence oscillates: xn = (−1)n(5N)12. The behavior for negative x0 is obtained by symmetry