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      <text>16.4.2 Radius of convergence question</text>
    </name>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the range of values of \(x\) for which the following power series converges: \[1 + \dfrac{x}{{@aa@}} + \dfrac{x^2}{{@aa@}^2} + \dfrac{x^3}{{@aa@}^3} + \cdots \]</p>
<p>First, find the coefficient of \(x^p\):</p>
<p>\(b_p=\) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>

<p> Now find \(R\), the radius of convergence:</p>
<p>\(R=\) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>

<p>Lastly, find what happens at \(x = \pm R\).</p>
<p>At \(x = R\), the series is: [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>
<p>At \(x = -R\), the series is: [[input:ans4]] [[validation:ans4]] [[feedback:prt4]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>We are given the series \(\displaystyle \sum_{p=1}^\infty \frac{x^p}{{@aa@}^p}\). We see then that the coefficient of \(x^p\) is \[b_p=\dfrac{1}{{@aa@}^p}.\]</p>

<p>To find \(R\), the radius of convergence, calculate \[R= \lim_{p \to \infty} \left| \dfrac{b_p}{b_{p+1}} \right| = \lim_{p \to \infty} \left|\dfrac{{@aa@}^{p+1}}{{@aa@}^p}\right| = \lim_{p \to \infty}{@aa@} = {@aa@}.\] When \(x=\pm {@aa@}\) the series
  is clearly <b>divergent</b> as the terms do not go to zero and we can apply the divergence test. Hence the series is convergent if and only if \(-{@aa@} &lt; x &lt; {@aa@}\).</p>]]></text>
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    <questionvariables>
      <text><![CDATA[aa:ev(rand(5)+2,simp);
ta1: 1/(aa^p);
ta2: aa;
ta3: [["divergent", true], ["convergent", false]];
ta4: [["divergent", true], ["convergent", false]];]]></text>
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    <questionnote format="moodle_auto_format">
      <text>\(1 + \dfrac{x}{{@aa@}} + \dfrac{x^2}{{@aa@}^2} + \dfrac{x^3}{{@aa@}^3} + \dots, \quad b_p= {@ta1@}, \quad R={@ta2@} \)</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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