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      <text>8.2.11 (Linearity)</text>
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      <text><![CDATA[<h3>Linearity</h3>

<p>One of the useful properties of sigma notation is that we can combine series using the property of linearity. Formally we have the following: </p>
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<p>If \(c\) is any constant (i.e. does not depend on \(n\)), then \[ \sum_{n=a}^{b} cx_n =c\sum_{n=a}^{b} x_n \] and \[ \sum_{n=a}^{b} (x_n+y_n)=\sum_{n=a}^{b} x_n+\sum_{n=a}^{b} y_n. \]</p>

<p>Another useful property (just like when we worked with definite integrals) is that we can split a sum up into parts:</p>
<p>If \(a,b,c\) are integers with \(a&lt;b&lt;c\) then \[\sum_{n=a}^{c} x_n=\sum_{n=a}^{b} x_n+\sum_{n=b+1}^{c} x_n.\]</p>]]></text>
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