Chapter 5 | Integration
509
interpreted as s 2 + s 3 + s 4 + s 5 + s 6 + s 7 . Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a dummy variable . We can use any letter we like for the index. Typically, mathematicians use i , j , k , m , and n for indices. Let’s try a couple of examples of using sigma notation. Example 5.1 Using Sigma Notation a. Write in sigma notation and evaluate the sum of terms 3 i for i =1,2,3,4,5. b. Write the sum in sigma notation: 1+ 1 4 + 1 9 + 1 16 + 1 25 . Solution a. Write ∑ i =1 5 3 i =3+3 2 +3 3 +3 4 +3 5 =363. b. The denominator of each term is a perfect square. Using sigma notation, this sum can be written as ∑ i =1 5 1 i 2 .
Write in sigma notation and evaluate the sum of terms 2 i for i =3, 4, 5, 6.
5.1
The properties associated with the summation process are given in the following rule.
Rule: Properties of Sigma Notation Let a 1 , a 2 ,…, a n and b 1 , b 2 ,…, b n represent two sequences of terms and let c be a constant. The following properties hold for all positive integers n and for integers m , with 1≤ m ≤ n . 1. (5.1) ∑ i =1 n c = nc 2. (5.2) ∑ i =1 n ca i = c ∑ i =1 n a i 3. (5.3) ∑ i =1 n ⎛ ⎝ a i + b i ⎞ ⎠ = ∑ i =1 n a i + ∑ i =1 n b i 4. (5.4) ∑ i =1 n ⎛ ⎝ a i − b i ⎞ ⎠ = ∑ i =1 n a i − ∑ i =1 n b i
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