Problem

Source: China south east mathematical Olympiad 2006 problem4

Tags: algebra unsolved, algebra



Given any positive integer $n$, let $a_n$ be the real root of equation $x^3+\dfrac{x}{n}=1$. Prove that (1) $a_{n+1}>a_n$; (2) $\sum_{i=1}^{n}\frac{1}{(i+1)^2a_i} <a_n$.