Show that $\frac{1}{\left( n-1 \right)!}-\frac{3}{n!}+\frac{2}{\left( n+1 \right)!}=\frac{A{{n}^{2}}

Question

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]

Accepted Answer

(i) $A=1$, $B=-2$, $C=-1$ (ii) $\dfrac15\left(\dfrac2{(N+1)!}-\dfrac1{N!}-1\right)$ (iii) Converges to $-\dfrac15$

2019 NYJC P2 Q1

2019 NYJC P2 Q1

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