Show that $\frac{1}{r!}-\frac{1}{\left( r+1 \right)!}=\frac{r}{\left( r+1 \right)!}$.[1] Henc

Question

Show that $\frac{1}{r!}-\frac{1}{\left( r+1 \right)!}=\frac{r}{\left( r+1 \right)!}$.[1] Hence find $\sum\limits_{r=1}^{N}{\frac{r}{\left( r+1 \right)!}}$ in terms of $N$.[3] Explain why the series in part (ii) is convergent, and hence state the value of $\sum\limits_{r=1}^{\infty }{\frac{r}{\left( r+1 \right)!}}$.[2] By using the result in part (ii), find $\sum\limits_{r=2}^{N}{\frac{r+2}{\left( r+3 \right)!}}$.[3]

Accepted Answer

(ii) $1-\frac{1}{\left( N+1 \right)!}$ (iii) $1$ (iv) $\frac{1}{24}-\frac{1}{\left( N+3 \right)!}$

2022 CJC Promo Q8

2022 CJC Promo Q8

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