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      <text>16.4.4 General power series</text>
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      <text><![CDATA[<h4>Power series centered at \(x_0\)&nbsp;</h4>
<p>A general power series has the form \[ b_0 + b_1(x-x_0) + b_2(x-x_0)^2 + \cdots = \sum_{p=0}^\infty b_p(x-x_0)^p\] </p>

<p> Exactly the same considerations apply to this general power series as apply to the ‘special’ series \(\sum_{p=0}^\infty b_p x^p\) except that the variable \(x\) is replaced by \((x-x_0).\) The radius of convergence of the general series is obtained in
    the same way: \[R = \lim_{p \to \infty} \left|\dfrac{b_p}{b_{p+1}}\right|\] and the interval of convergence is now shifted to have centre at \(x = x_0\) (see Figure below). The series is absolutely convergent if \(|x − x_0| &lt; R\), diverges if \(|x
    − x_0| &gt; R\) and may or may not converge if \(|x − x_0| = R\).</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/16-4-fig-11-Convergence.svg" alt="Interval of convergence" width="456" height="55" class="img-responsive atto_image_button_text-bottom"></figure>]]></text>
<file name="16-4-fig-11-Convergence.svg" path="/" encoding="base64">[encoded data, 43372 characters]</file>
<file name="fig4.png" path="/" encoding="base64">[encoded data, 31600 characters]</file>
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