<p>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 $n\ge 2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\sum\limits_{r=2}^{n}{\frac{{{r}^{2}}-r-1}{r!}}=2-\frac{n+1}{n!}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find $\sum\limits_{r=8}^{n+5}{\frac{{{r}^{2}}-3r+1}{\left( r-1 \right)!}}$ in terms of $n$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>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 $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}>0$, then both $\sum\limits_{r=k}^{\infty }{{{a}_{r}}}$and $\sum\limits_{r=k}^{\infty }{{{b}_{r}}}$ converges or both diverges.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">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.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that $e-2<\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...<2$.</div><p style="text-align: right;">[3]</p></li></ol>

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 $n\ge 2$.

Show that $\sum\limits_{r=2}^{n}{\frac{{{r}^{2}}-r-1}{r!}}=2-\frac{n+1}{n!}$.

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

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 $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}>0$, then both $\sum\limits_{r=k}^{\infty }{{{a}_{r}}}$and $\sum\limits_{r=k}^{\infty }{{{b}_{r}}}$ converges or both diverges.

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.

Show that $e-2<\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...<2$.

Techniques of Differentiation - General

Techniques of Differentiation

H2 Math

International Baccalaureate Higher Level Mathematics

2018 SRJC P1 Q8

2018 SRJC P1 Q8

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