2022 CJC Promo Q8
(i)
Show that $\frac{1}{r!}-\frac{1}{\left( r+1 \right)!}=\frac{r}{\left( r+1 \right)!}$.
[1]
(ii)
Hence find $\sum\limits_{r=1}^{N}{\frac{r}{\left( r+1 \right)!}}$ in terms of $N$.
[3]
(iii)
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]
(iv)
By using the result in part (ii), find $\sum\limits_{r=2}^{N}{\frac{r+2}{\left( r+3 \right)!}}$.
[3]
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