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      <text>23.3.2.4 Confirm Fourier series result</text>
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    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>Confirm the result obtained for the triangular wave, function 7 in the last Task,
by finding the Fourier series fully. The function involved is</p>
<p> \[\begin{align*} f(t) & = |t| \quad\quad - \pi < t < \pi, \\ f(t+2\pi) & = f(t). \end{align*}\]</p>
<h4 class="HELM_solution">Solution</h4>
<p>Since \(f(t)\) is even we can say immediately \[b_n=0\qquad n=1,2,3,\ldots\] Also \[\begin{aligned}
a_n=\frac{2}{\pi}\int_0^\pi t\cos nt \, dt=\left\{\begin{array}{ccc}0&amp;&amp;n\ \text{even}\\ \\ -\frac{4}{n^2\pi}&amp;&amp;n\ \text{odd}\end{array}\right.
\end{aligned}\] (after integration by parts). Also \(a_0=\frac{2}{\pi}\displaystyle\int_0^\pi t\ d t=\pi\) so the Fourier series is \[f(t)=\frac{\pi}{2}-\frac{4}{\pi}\left(\cos t+\frac{1}{9}\cos 3t+\frac{1}{25}\cos
5t+\ldots \right)\]</p>]]></text>
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