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      <text>23.4.1.1 Convergence of a Fourier series</text>
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      <text><![CDATA[<h4>Convergence of a Fourier series</h4>
We have now shown how to obtain a Fourier series for periodic functions. We have suggested that
we would expect to be able to approximate such functions by using a few terms of the Fourier series.
The detailed question of the <b>convergence</b> or otherwise of Fourier series has not been discussed.
The reason for this is that the great majority of functions likely to be encountered in practice have
Fourier series that do indeed converge and can therefore be safely used as approximations.
<p></p>
The precise conditions that have to be fulfilled for a Fourier series to converge are known as Dirichlet
conditions after the French mathematician who investigated the matter. The three conditions are
listed in the following Key Point.
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<h4>Key Point</h4>
<p>The Dirichlet conditions for the convergence of a Fourier series of a periodic function \(f(t)\) are:</p>
<p></p>
1. \(f(t)\) must have only a finite number of finite discontinuities, within one period
<p></p>
2. \(f(t)\) must have a finite number of maxima and minima over one period
<p></p>
3. the integral \( \displaystyle{ \int _{-\frac{T}{2}}^{\frac{T}{2}} } \lvert f(t) \rvert dt \) must be finite
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