2022 HCI P1 Q3

It is given that \({{u}_{r}}=\frac{1}{r!}\), \(r\in {{\mathbb{Z}}^{+}}\).

1. Show that \({{u}_{r}}-2{{u}_{r+1}}+{{u}_{r+2}}=\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!}\).[1]
2. Hence find \(\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!}}\) in terms of \(n\) and determine the value of \(\sum\limits_{r=1}^{\infty }{\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!}}\).[3]
3. Using an expansion from the List of Formulae MF26 and the answer in part (ii), find the exact value of \(\sum\limits_{r=1}^{\infty }{\left( {{u}_{r}}-\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!} \right)}\).[2]