2021 NYJC Promo Q5

By writing \(\frac{2-r}{r\left( r+1 \right)\left( r+2 \right)}\) in partial fractions, show that \(\sum\limits_{r=1}^{n}{\frac{2-r}{r\left( r+1 \right)\left( r+2 \right)}=\frac{An}{\left( n+1 \right)\left( n+2 \right)}}\), where \(A\) is a constant to be determined.[4]

Using the result in part (i), find \(\sum\limits_{r=0}^{n}{\frac{1-r}{\left( r+1 \right)\left( r+2 \right)\left( r+3 \right)}}\) in terms of \(n\).[2]

Hence find the exact value of \(\sum\limits_{r=10}^{\infty }{\frac{1-r}{\left( r+1 \right)\left( r+2 \right)\left( r+3 \right)}}\).[2]