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      <text>16.1.3.1 Series</text>
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      <text><![CDATA[<h3>Finite Series</h3>
<p>A <strong>series</strong> is the sum of the terms of a sequence. For example, the <strong>harmonic series</strong> is \[ 1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\cdots\]and the <strong>alternating harmonic series</strong> is \[ 1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\cdots.\]</p>We will mainly consider finite series in this section, that is, series which are the sum of the first \(k\) terms of a sequence for some natural number \(k\).<br><br>

<h4>The summation notation</h4>
<p>If we consider a general sequence \[ a_1,\ a_2,\ \ldots ,\ a_n,\ \ldots\] then the sum of the first \(k\) terms \(a_1+a_2+a_3+\cdots +a_k\) is concisely denoted by \(\displaystyle \sum_{p=1}^k\ a_p\).</p>
<p>That is, \[ a_1+a_2+a_3+\cdots +a_k= \sum_{p=1}^ka_p.\]</p> 

<p>When we encounter the expression \(\displaystyle \sum_{p=1}^k a_p\) we let the index ‘\(p\)’ in the term \(a_p\) take, in turn, the values \(1,2,\ldots,k\) and then add all these terms together. So, for example \[ \sum_{p=1}^3a_p= a_1+a_2+a_3\qquad\qquad \sum_{p=2}^7a_p= a_2+a_3+a_4+a_5+a_6+a_7\] Note that \(p\) is a <strong>dummy</strong> index; any letter could be used as the index. For example \(\displaystyle \sum_{i=1}^6 a_i\), and \(\displaystyle \sum_{m=1}^6 a_m\) each represent the same collection of terms: \(a_1+a_2+a_3+a_4+a_5+a_6\).</p>
<p>In order to be able to use this ‘summation notation’ we need to obtain a suitable expression for the ‘typical term’ in the series. For example, the finite series \[ 1^2+2^2+\cdots +k^2\]may be written as \(\displaystyle \sum_{p=1}^k p^2\) since the typical term is clearly \(p^2\) in which \(p=1,2,3,\ldots, k\) in turn.</p>

<p>In the same way \[ 1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\cdots -\dfrac{1}{16}= \sum_{p=1}^{16}\dfrac{(-1)^{p+1}}{p}\]since an expression for the typical term in this alternating harmonic series is \(a_p= \dfrac{(-1)^{p+1}}{p}\).<br><br>Note that we are using a convention that the terms of the series start with index \(k=1\). When working with power series we will more often see starting index \(k=0\). It is very important to pay attention to this value when summing a series.</p>]]></text>
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