For every positive integer $ n$, prove the inequality: $ \frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+...+\frac{n+2}{n!+(n+1)!+(n+2)!}<\frac{1}{2}.$
Source: Ukraine 2005 grade 9
Tags: inequalities, inequalities proposed
For every positive integer $ n$, prove the inequality: $ \frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+...+\frac{n+2}{n!+(n+1)!+(n+2)!}<\frac{1}{2}.$