2015 HCI Promo Q3

Prove that \(\frac{3{{r}^{2}}-2r-3}{\left( r+1 \right)!}=\frac{3}{\left( r-1 \right)!}-\frac{5}{r!}+\frac{2}{\left( r+1 \right)!}\), showing your workings clearly.[1]

Hence evaluate \(\sum\limits_{r=1}^{n}{\frac{3{{r}^{2}}-2r-3}{\left( r+1 \right)!}}\), express your answer in the form \(1-\mathrm{f}\left( n \right)\) where \(\mathrm{f}\left( n \right)\) is a single fraction in terms of \(n\).[3]

Deduce that \(\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}-r-1}{\left( r+1 \right)!}}<\frac{1}{3}\) .[2]