2020 MI P1 Q3

A sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), ... is given by

\({{u}_{r}}=\frac{1}{\left( r+1 \right)!}\) for \(r\ge 1\).

1. Show that \({{u}_{r}}-{{u}_{r+1}}=\frac{r+1}{\left( r+2 \right)!}\).[1]

It is given that \({{S}_{n}}=\sum\limits_{r=1}^{n}{\frac{r+1}{\left( r+2 \right)!}}\).

2. Hence find an expression for \({{S}_{n}}\) in terms of \(n\). (There is no need to express your answer as a single algebraic fraction.)[4]
3. Find the smallest value of \(n\) for which \({{S}_{n}}\) is within \({{10}^{-4}}\) of the sum to infrnity.[3]