Problem #PRU-61311

Problems Calculus Number sequences Limit of a sequence, convergence Algebra and arithmetic 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: \(a_0 = a > 0\), \(a_{n + 1} = 1/3 (2a_n + a/a^2_n)\) (\(n \geq 0\)).

Prove that \(\lim\limits_{n\to\infty} a_n = \sqrt[3]{a}\).