Show that $\frac{1}{(r-1)!} - \frac{2}{r!} + \frac{1}{(r+1)!} = \frac{\mathrm{f}(r)}{(r+1)!}$

Question

Show that $\frac{1}{(r-1)!} - \frac{2}{r!} + \frac{1}{(r+1)!} = \frac{\mathrm{f}(r)}{(r+1)!}$, where $\mathrm{f}(r)$ is a function in $r$ to be found.[1] The sum $\sum_{r=2}^{N} \frac{\mathrm{f}(r)}{(r+1)!}$ is denoted by $S_N$. Using your answer in part (a), find $S_N$ in terms of $N$.[3] Give a reason why $S_N$ converges and find the exact value of $S_{\infty}$.[2] Find the smallest value of $N$ such that $S_N$ is within $10^{-7}$ of $S_{\infty}$.[2]

Accepted Answer

(a) ${\mathrm{f}(r)=r^2-r-1}$ (b) ${\frac{1}{2}-\frac{N}{(N+1)!}}$ (c) ${S_{\infty}=\frac{1}{2}}$ (d) $11$

2022 YIJC P1 Q5

2022 YIJC P1 Q5