2018 SRJC P1 Q8

A sequence \({{u}_{0}}\), \({{u}_{1}}\), \({{u}_{2}}\)… is such that \({{u}_{r}}=\frac{1}{r!}\) and \({{u}_{r}}={{u}_{r-2}}+\frac{r+1-{{r}^{2}}}{r!}\), when \(n\ge 2\).

1. Show that \(\sum\limits_{r=2}^{n}{\frac{{{r}^{2}}-r-1}{r!}}=2-\frac{n+1}{n!}\).[3]
2. Hence find \(\sum\limits_{r=8}^{n+5}{\frac{{{r}^{2}}-3r+1}{\left( r-1 \right)!}}\) in terms of \(n\).[3]

Limit Comparison test states that for two series of the form \(\sum\limits_{r=k}^{n}{{{a}_{r}}}\) and \(\sum\limits_{r=k}^{n}{{{b}_{r}}}\) with \({{a}_{n}},{{b}_{n}}\ge 0\) for all \(n\), if \(\mathop{\lim}\limits_{n\to \infty }\,\frac{{{a}_{n}}}{{{b}_{n}}}>0\), then both \(\sum\limits_{r=k}^{\infty }{{{a}_{r}}}\)and \(\sum\limits_{r=k}^{\infty }{{{b}_{r}}}\) converges or both diverges.

3. Given that \(\sum\limits_{r=2}^{\infty }{\frac{{{r}^{2}}-r-1}{r!}}\) is convergent, using the test, explain why \(\sum\limits_{r=2}^{\infty }{\frac{r-2}{\left( r-1 \right)!}}\) is convergent.[2]
4. Show that \(e-2<\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...<2\).[3]