2021 TJC P2 Q2

It is given that \(\mathrm{f}\left( r \right)=\frac{1}{r!}\) where \(r\) is a positive integer.

1. Show that \(\mathrm{f}\left( r \right)-\mathrm{f}\left( r+1 \right)=\frac{r}{\left( r+1 \right)!}\).[1]
2. The sum of the first \(n\) terms of the series \(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...\) is given by \({{S}_{n}}\). Find \({{S}_{n}}\) and explain why the series converges.[4]
3. By considering (ii) and using the standard series from the List of Formulae (MF26), find the exact value of \(\sum\limits_{r=2}^{\infty }{\frac{r+2}{r!}}\).[4]