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    <name>
      <text>8.3.14 (Example of geometric and p series convergence)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h3>Example</h3>

<p>Use the series tests above to decide if the following series are convergent or divergent.</p>
<ol type="a">
<li>\(\sum\limits_{n=1}^\infty \frac{1}{8n^3}\).</li>
<li>\(\sum\limits_{n=1}^\infty 24 \cdot 2^{n-1}\)</li>
</ol>

<h3>Solution</h3>

<ol type="a">
<li>In this case we note that 
\[
\frac{1}{8n^3} = \frac{1}{(2n)^3},
\]
so this is a p-series with \(p=3 > 1\) and therefore this series is convergent.</li>
<li>This series is a geometric series with \(r = 2\), therefore it diverges.</li>
</ol>]]></text>
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