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Math 151 Section 6.1 Summation Notation Definition: A sequence of real numbers is a function, range the real numbers. We usually write Example: f (n) 1 an n f (n ) , with domain the nonnegative integers and f ( n ) an . We often write the sequence as an ordered list, listing enough terms to see the formula and then … . For this example, we could write the sequence as 1 1 1 1, , , , ... 2 3 4 When we want to add many terms of a given sequence, we use summation notation. The Greek letter sigma, ∑, is used for summations. 5 1 1 1 1 1 Example: 1 2 3 4 5 n 1 n n is called the index in the sum. Examples: Evaluate each sum. 9 1. n n4 4 2. 1 k 1 k 2 Reindexing: If the index 2 1 k 1 k 2 8 7 k varies from 2 to 8 then k 1 j varies from 1 to 7. We can write, 1 2 j 1 j . All the usual rules of arithmetic apply to finite sums. 5 1 5 3 7 k 3 7 k k k k 2 k 2 k 2 5 Example: Two formulas for adding certain sums: n ( n 1) k 2 k 1 n n 2 k k 1 n ( n 1)( 2 n 1) 6 Examples: 4 1. i ( i 3) i 1 5 2 k 3k 100 2. k 0 2 1 1 2 k 12 k 1 k 10 3. Example 3 is called a telescoping or collapsing sum.