Problem #PRU-61311

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

Problem

The algorithm of the approximate calculation of $\sqrt[3]{a}$ . The sequence ${a_{n}}$ is defined by the following conditions: $a_{0} = a > 0$ , $a_{n + 1} = 1 / 3 (2 a_{n} + a / a_{n}^{2})$ ( $n \geq 0$ ).

Prove that $lim_{n \to \infty} a_{n} = \sqrt[3]{a}$ .

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