2018 NYJC Promo Q6

Show that \(\frac{1}{\left( r-1 \right)r\left( r+1 \right)}\) can be expressed as \(\frac{A}{r-1}+\frac{B}{r}+\frac{C}{r+1}\), where \(A\), \(B\) and \(C\) are constants to be determined.[1]

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

Explain why the series \(\sum\limits_{r=2}^{\infty }{\frac{2}{\left( r-1 \right)r\left( r+1 \right)}}\) converges.[1]

By using (ii) or otherwise, evaluate \(\sum\limits_{r=1}^{N-1}{\frac{r!}{\left( r+3 \right)!}}\), leaving your answer in terms of \(N\).[4]