2022 HCI P1 Q3
It is given that ${{u}_{r}}=\frac{1}{r!}$, $r\in {{\mathbb{Z}}^{+}}$.
(i)
Show that ${{u}_{r}}-2{{u}_{r+1}}+{{u}_{r+2}}=\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!}$.
[1]
(ii)
Hence find $\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!}}$ in terms of $n$ and determine the value of $\sum\limits_{r=1}^{\infty }{\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!}}$.
[3]
(iii)
Using an expansion from the List of Formulae MF26 and the answer in part (ii), find the exact value of $\sum\limits_{r=1}^{\infty }{\left( {{u}_{r}}-\frac{{{r}^{2}}+r-1}{\left( r+2 \right)!} \right)}$.
[2]
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