2019 NYJC P2 Q1

Show that $\frac{1}{\left( n-1 \right)!}-\frac{3}{n!}+\frac{2}{\left( n+1 \right)!}=\frac{A{{n}^{2}}+Bn+C}{\left( n+1 \right)!}$ where $A$, $B$ and $C$ are constants to be determined.

[2]

Hence find $\sum\limits_{n=1}^{N}{\frac{{{n}^{2}}-2n-1}{5\left( n+1 \right)!}}$ in terms of $N$.

[3]

Give a reason why the series $\sum\limits_{n=1}^{\infty }{\frac{{{n}^{2}}-2n-1}{5\left( n+1 \right)!}}$ converges and write down its value.

[2]