A recurrence relation for the square root |
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| Authors: | W. G. Hwang John Todd |
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| Affiliation: | Center for Applied Mathematics, Cornell University, Ithaca, New York 14850 U.S.A.;Department of Mathematics, California Institute of Technology, Pasadena, California 91109 U.S.A. |
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| Abstract: | The behavior of the sequence xn + 1 = xn(3N − xn2)/2N is studied for N > 0 and varying real x0. When 0 < x0 < (3N)1/2 the sequence converges quadratically to N1/2. When x0 > (5N)1/2 the sequence oscillates infinitely. There is an increasing sequence βr, with β−1 = (3N)1/2 which converges to (5N)1/2 and is such that when βr < x0 < βr + 1 the sequence {xn} converges to (−1)rN1/2. For x0 = 0, β−1, β0,… the sequence converges to 0. For x0 = (5N)1/2 the sequence oscillates: xn = (−1)n(5N)1/2. The behavior for negative x0 is obtained by symmetry. |
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