2018 YJC Promo Q13

Show that \(\frac{3}{r}-\frac{6}{r+1}+\frac{3}{r+2}=\frac{6}{r\left( r+1 \right)\left( r+2 \right)}\).[1]

Hence show that \(\sum\limits_{r=1}^{N}{\frac{1}{r\left( r+1 \right)\left( r+2 \right)}}=\frac{1}{4}-\frac{1}{2\left( N+1 \right)}+\frac{1}{2\left( N+2 \right)}\).[3]

Give a reason why the series \(\sum\limits_{r=1}^{\infty }{\frac{1}{r\left( r+1 \right)\left( r+2 \right)}}\) converges and write down its value.[2]

Use your answer to (ii) to find \(\sum\limits_{r=3}^{N}{\frac{1}{r\left( r-1 \right)\left( r-2 \right)}}\)in terms of \(N\).[2]