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      <text>16.5.4.1 The Taylor series</text>
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      <text><![CDATA[<h3><span>The Taylor series</span></h3><br>
<p>The <strong>Taylor series</strong> is a generalisation of the Maclaurin series being a power series developed in powers of <span class="math inline">\((x-x_0)\)</span> rather than in powers of <span class="math inline">\(x\)</span>.</p>
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    <h4>Key Point</h4>
    <p>If the function <span class="math inline">\(f(x)\)</span> can be differentiated as often as required at <span class="math inline">\(x=x_0\)</span> then: <span class="math display">\[f(x)=f(x_0)+(x-x_0)f'(x_0)+\dfrac{(x-x_0)^2}{2!}f''(x_0)+\cdots\]</span>        This is called the Taylor series of <span class="math inline">\(f(x)\)</span> about the point <span class="math inline">\(x_0\)</span>.</p>
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<p>The Maclaurin expansion is the Taylor expansion obtained if <span class="math inline">\(x_0\)</span> is chosen to be zero.</p>]]></text>
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