2018 NYJC Promo Q6
(i)
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]
(ii)
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]
(iii)
Explain why the series $\sum\limits_{r=2}^{\infty }{\frac{2}{\left( r-1 \right)r\left( r+1 \right)}}$ converges.
[1]
(iv)
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]
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