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    <name>
      <text>16.5.2.3 Find the Maclaurin series</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the Maclaurin expansion of <span class="math inline">\(\ln(1+x)\)</span>. (Note that we <span><strong>cannot</strong></span> find a Maclaurin expansion of the function <span class="math inline">\(\ln x\)</span> since <span class="math inline">\(\ln x\)</span>    does not exist at <span class="math inline">\(x=0\)</span> and so cannot be differentiated at <span class="math inline">\(x=0\)</span>.)</p>
<p>Find the first four derivatives of <span class="math inline">\(f(x)=\ln(1+x)\)</span>:</p>
<p>Now obtain </p>
<ol class="HELM_parts">

    <li>
        <p>\(f(0) =\) [[input:ans1]] [[validation:ans1]]&nbsp;[[feedback:prt1]]</p>
    </li>

    <li>
        <p>\(f'(0) =\) [[input:ans2]] [[validation:ans2]]&nbsp;[[feedback:prt2]]</p>
    </li>
    <li>
        <p>\(f''(0) =\) [[input:ans3]] [[validation:ans3]]&nbsp;[[feedback:prt3]]</p>
    </li>
    <li>
        <p>\(f'''(0) =\) [[input:ans4]] [[validation:ans4]]&nbsp;[[feedback:prt4]]</p>
    </li>

</ol>
<p></p><p>
    Hence, obtain the Maclaurin expansion of \(\ln (1+x)\): (<em>Hint: enter the general term for \(x^n\)</em>)</p>
<p>
    \(\ln(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\cdots +\)[[input:ans5]][[validation:ans5]] \(+\cdots \) [[feedback:prt5]]
</p>

<p>Now obtain the radius of convergence.</p>

<p>\(R = \) [[input:ans6]] [[validation:ans6]] [[feedback:prt6]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>Find the first four derivatives of \(f(x)=\ln(1+x)\):</p>
<p>\(f'(x)=\dfrac{1}{1+x},\qquad f''(x)=\dfrac{-1}{(1+x)^2},\qquad f'''(x)=\dfrac{2}{(1+x)^3}\).<br></p>
<p>Generally \( f^{(n)}(x)=\dfrac{(-1)^{n+1}(n-1)!}{(1+x)^n}.\)</p>
<p><br></p>
<p>Now obtain \(f(0)=0 \qquad f'(0)=1,\qquad f''(0)=-1,\qquad f'''(0)=2\).</p>
<p>Generally \(f^{(n)}(0)=(-1)^{n+1}(n-1)!\).</p>
<p><br></p>
<p>Hence the Maclaurin expansion of \[\ln (1+x) \) is \( \ln(1+x) = x-\dfrac{x^2}{2}+\dfrac{x^3}{3} + \cdots +\dfrac{(-1)^{n+1}}{n}x^n+\cdots\]
</p><p>Now the radius of convergence is \(R=1\).</p>]]></text>
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      <text>f:0;
fprime1: 1;
fprime2:-1;
fprime3: 2;
fgeneral: ((-1)^(n+1)* x^n)/n
radius:1</text>
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    <questionnote format="moodle_auto_format">
      <text>Find the Maclaurin expansion of \(\ln(1+x)\)</text>
    </questionnote>
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      <text/>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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