Hykilpikonna
120 lines
7.5 KiB
HTML
120 lines
7.5 KiB
HTML
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<header id="title-block-header">
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<h1 class="title">C.1 Summations and Products</h1>
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<section>
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<p>When performing calculations, we’ll often end up writing sums of terms, where each term follows a pattern. For example: <span class="math display">\[\frac{1 + 1^2}{3 + 1} +
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\frac{2 + 2^2}{3 + 2} +
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\frac{3 + 3^2}{3 + 3} +
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\cdots +
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\frac{100 + 100^2}{3 + 100}\]</span></p>
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<p>We will often use <em>summation notation</em> to express such sums concisely. We could rewrite the previous example simply as: <span class="math display">\[\sum_{i=1}^{100} \frac{i + i^2}{3 + i}.\]</span></p>
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<p>In this example, <span class="math inline">\(i\)</span> is called the <em>index of summation</em>, and <span class="math inline">\(1\)</span> and <span class="math inline">\(100\)</span> are the <em>lower</em> and <em>upper bounds</em> of the summation, respectively. A bit more generally, for any pair of integers <span class="math inline">\(j\)</span> and <span class="math inline">\(k\)</span>, and any function <span class="math inline">\(f : \Z \to \R\)</span>, we can use summation notation in the following way: <span class="math display">\[\sum_{i=j}^k f(i) = f(j) + f(j+1) + f(j+2) + \dots + f(k).\]</span></p>
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<p>We can similarly use <em>product notation</em> to abbreviate multiplication:<label for="sn-0" class="margin-toggle sidenote-number"></label><input type="checkbox" id="sn-0" class="margin-toggle"/><span class="sidenote">Fun fact: the Greek letter <span class="math inline">\(\Sigma\)</span> (sigma) corresponds to the first letter of “sum,” and the Greek letter <span class="math inline">\(\Pi\)</span> (pi) corresponds to the first letter of “product.”</span> <span class="math display">\[\prod_{i=j}^k f(i) = f(j) \times f(j+1) \times f(j+2) \times \dots \times f(k).\]</span></p>
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<p>It is sometimes useful (e.g., in certain formulas) to allow a summation or product’s lower bound to be greater than its upper bound. In this case, we say the summation or product is <em>empty</em>, and define their values as follows:<label for="sn-1" class="margin-toggle sidenote-number"></label><input type="checkbox" id="sn-1" class="margin-toggle"/><span class="sidenote">These particular values are chosen so that adding an empty summation and multiplying by an empty product do not change the value of an expression.</span></p>
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<ul>
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<li>When <span class="math inline">\(j > k\)</span>, <span class="math inline">\(\sum_{i=j}^k f(i) = 0\)</span>.</li>
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<li>When <span class="math inline">\(j > k\)</span>, <span class="math inline">\(\prod_{i=j}^k f(i) = 1\)</span>.</li>
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</ul>
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<p>Finally, we’ll end off this section with a few formulas for common summation formulas, and a few laws governing how expressions using summation and product notation can be simplified.</p>
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<div id="theorem:summation_formulas" class="theorem">
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<p>For all <span class="math inline">\(n \in \N\)</span>, the following formulas hold:</p>
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<ol type="1">
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<li>For all <span class="math inline">\(c \in \R\)</span>, <span class="math inline">\(\sum_{i=1}^{n} c = c \cdot n\)</span> (sum with constant terms).</li>
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<li><span class="math inline">\(\sum_{i=0}^{n} i = \frac{n(n+1)}{2}\)</span> (sum of consecutive numbers).</li>
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<li><span class="math inline">\(\sum_{i=0}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\)</span> (sum of consecutive squares).</li>
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<li>For all <span class="math inline">\(r \in \R\)</span>, if <span class="math inline">\(r \neq 1\)</span> then <span class="math inline">\(\sum_{i=0}^{n-1} r^i = \frac{r^n - 1}{r - 1}\)</span> (sum of powers).</li>
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<li>For all <span class="math inline">\(r \in \R\)</span>, if <span class="math inline">\(r \neq 1\)</span> then <span class="math inline">\(\sum_{i=0}^{n-1} i \cdot r^i = \frac{n \cdot r^n}{r - 1} - \frac{r(r^n - 1)}{(r - 1)^2}\)</span> (arithmetico-geometric series).</li>
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</ol>
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</div>
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<div id="theorem:summation_product_laws" class="theorem">
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<p>For all <span class="math inline">\(m, n \in \Z\)</span>, the following formulas hold:</p>
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<ol type="1">
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<li><p><span class="math inline">\(\sum_{i=m}^{n} (a_i + b_i) = \left( \sum_{i=m}^{n} a_i \right) + \left(\sum_{i=m}^{n} b_i \right)\)</span> (separating sums)</p></li>
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<li><p><span class="math inline">\(\prod_{i=m}^{n} (a_i \cdot b_i) = \left( \prod_{i=m}^{n} a_i \right) \cdot \left (\prod_{i=m}^{n} b_i \right)\)</span> (separating products)</p></li>
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<li><p><span class="math inline">\(\sum_{i=m}^{n} c \cdot a_i = c \cdot \left( \sum_{i=m}^{n} a_i \right)\)</span> (factoring out constants, sums)</p></li>
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<li><p><span class="math inline">\(\prod_{i=m}^{n} c \cdot a_i = c^{n - m + 1} \cdot \left( \prod_{i=m}^{n} a_i \right)\)</span> (factoring out constants, products)</p></li>
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<li><p><span class="math inline">\(\sum_{i=m}^{n} a_i = \sum_{i'=0}^{n-m} a_{i'+m}\)</span> (change of index <span class="math inline">\(i' = i - m\)</span>)</p></li>
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<li><p><span class="math inline">\(\prod_{i=m}^{n} a_i = \prod_{i'=0}^{n-m} a_{i'+m}\)</span> (change of index <span class="math inline">\(i' = i - m\)</span>)</p></li>
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</ol>
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</section>
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<a href="https://www.teach.cs.toronto.edu/~csc110y/fall/notes/">CSC110 Course Notes Home</a>
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