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Sequences Which Converge to Square Roots and Cube Roots

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Use the Secant method (defined in the book) to show that sequence below converges to (square root Q), where Q > 0, given "good" starting values x_0 and x_1:

x_n+1 = (x_n x_n-1 + Q) / (x_n + x_n-1).

Come up with a similar recursion for calculating Q^(1/3) using the secant method.

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Note that if the sequence converges to a positive value, it must converge to sqrt(Q). Suppose ...

Solution Summary

We show that a given system of two real-valued sequences converges to the square root of the first number in the first sequence and show how to derive a system of sequences which converge to the cube root of the first number in the first sequence.