Sums and Products
An important shorthand notation for calculations.
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Study Sums and Products
We will use the following notations. \[\begin{aligned} \sum_{i=1}^n a_i &= a_1+a_2+\dots+a_{n-1}+a_n\\ \prod_{i=1}^n a_i &= a_1 \cdot a_2 \cdot \dots \cdot a_{n-1}\cdot a_n\\ \bigcup_{i=1}^n A_i &= A_1 \cup A_2 \cup \dots \cup A_{n-1}\cup A_n \end{aligned}\] The union works also for an arbitrary index set \(\mathcal{I}\): \[\bigcup_{i \in \mathcal{I}} A_i = \{ x \,:\, \exists i \in \mathcal{I} \text{ with } x \in A_i \} \,.\]
The first is a useful notation for a sum which is the result
of an addition. Two or more summands added. Instead of using
points, we use the Greek letter \(\sum\). For example, \[3+7+15+\ldots+127\] is not an unambiguous
way to describe the sum. Using the sum symbol, there is no confusion:
\[%\label{eq:sum1}
\sum_{i=2}^7 (2^i-1).\] Of course, the parentheses are necessary
here. You can read this as a for loop:
Concept 1 (for loop for the sum above). sum := 0;
for i:=2 to 7 do {
sum := sum + 2^i-1);
}
Rule of Thumb 2. Let \(i\) run from \(2\) to \(7\), calculate \(2^i-1\) and add.
| index variable: | \(i=2\), | first summand: | \(2^i-1=2^2-1=\) | \(4-1=\) | \(3\) |
| index variable: | \(i=3\), | second summand: | \(2^i-1=2^3-1=\) | \(8-1=\) | \(7\) |
| index variable: | \(i=4\), | third summand: | \(2^i-1=2^4-1=\) | \(16-1=\) | \(15\) |
| index variable: | \(i=5\), | fourth summand: | \(2^i-1=2^5-1=\) | \(32-1=\) | \(31\) |
| index variable: | \(i=6\), | fifth summand: | \(2^i-1=2^6-1=\) | \(64-1=\) | \(63\) |
| index variable: | \(i=7\), | last summand: | \(2^i-1=2^7-1=\) | \(128-1=\) | \(127\) |
| Sum: | \(246\) |
Example 3. \[\sum\limits_{i=1}^{10} (2i-1) = 1+3+5+\ldots+19 \stackrel?= 100\] \[\sum\limits_{i=-10}^{10} i = -10-9-8-\ldots-1+0+1+\dots+8+9+10 \stackrel?=0\]
With the same construction, we describe the result of a multiplication, called a product, which consists of two or more factors. There we use the Greek letter \(\prod\). For example: \[\prod_{i=1}^8 (2i) = (2\cdot 1)\cdot (2\cdot 2)\cdot (2\cdot 3)\cdot \ldots \cdot (2\cdot 8) \stackrel?= 10321920.\]
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