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      <text>8.3.15 (Example using divergence test)</text>
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      <text><![CDATA[<h3>Example</h3>

<p>Do the following series converge or diverge?
<ol type="a">
<li>\(\sum\limits_{n=1}^\infty \frac{n^2 - 1}{n^2+n}\)</li>
<li>\(\sum\limits_{n=1}^\infty \frac{n - 1}{n^2+n}\)</li>
</ol>

<h3>Solution</h3>

<ol type="a">
<li>We use the divergence test. Here we have 
\[
\lim\limits_{n \to \infty} \frac{n^2 - 1}{n^2+n} = \lim\limits_{n \to \infty} \frac{1-\frac{1}{n^2}}{1+\frac{1}{n}}=1 \not=0. 
\]
Therefore by the divergence test the series diverges.</li>
<li>We proceed similarly with this series
\[
\lim\limits_{n \to \infty} \frac{n - 1}{n^2+n} = \lim\limits_{n \to \infty} \frac{1 - \frac{1}{n}}{n + 1} = \lim\limits_{n \to \infty} \frac{1}{n+1} \to 0 \mbox{ as } n \to \infty.
\]
Therefore the series passes the divergence test but notice that this does <b>not</b> tell us that the initial series converges. It tells us that we need more information to decide.</li>
</ol>]]></text>
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