Problem #PRU-61338

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Problem

The sequence of numbers $a_{1}, a_{2}, a_{3}, \dots$ is given by the following conditions $a_{1} = 1$ , $a_{n + 1} = a_{n} + \frac{1}{a_{n}^{2}}$ ( $n \geq 0$ ).

Prove that

a) this sequence is unbounded;

b) $a_{9000} > 30$ ;

c) find the limit $lim_{n \to \infty} \frac{a_{n}}{\sqrt[3]{n}}$ .

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