<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, ... is given by</p>
<p style="text-align: center">${{u}_{r}}=\frac{1}{\left( r+1 \right)!}$ for $r\ge 1$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that ${{u}_{r}}-{{u}_{r+1}}=\frac{r+1}{\left( r+2 \right)!}$.</span></div><p style="text-align: right;">[1]</p></li></ol>
<p>It is given that ${{S}_{n}}=\sum\limits_{r=1}^{n}{\frac{r+1}{\left( r+2 \right)!}}$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Hence find an expression for ${{S}_{n}}$ in terms of $n$. (There is no need to express your answer as a single algebraic fraction.)</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the smallest value of $n$ for which ${{S}_{n}}$ is within ${{10}^{-4}}$ of the sum to infrnity.</span></div><p style="text-align: right;">[3]</p></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, ... is given by

${{u}_{r}}=\frac{1}{\left( r+1 \right)!}$ for $r\ge 1$.

Show that ${{u}_{r}}-{{u}_{r+1}}=\frac{r+1}{\left( r+2 \right)!}$.

It is given that ${{S}_{n}}=\sum\limits_{r=1}^{n}{\frac{r+1}{\left( r+2 \right)!}}$.

Hence find an expression for ${{S}_{n}}$ in terms of $n$. (There is no need to express your answer as a single algebraic fraction.)

Find the smallest value of $n$ for which ${{S}_{n}}$ is within ${{10}^{-4}}$ of the sum to infrnity.

Recurrence Relations and Summation

International Baccalaureate Higher Level Mathematics

2020 MI P1 Q3

2020 MI P1 Q3

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