1. Section 13.6: Sigma Notation

2. ∑ The Greek letter, sigma, shown above, is very often used in mathematics to represent the sum of a series.

3. Is shorthand for the series starting with the first term and ending with the ninth term of 3k. That is as follows: = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6) + 3(7) + 3(8) + 3(9)

4. The symbol 3k is called the summand. The numbers 1 and 9 are the limits of the summation. The symbol k is the index. The choice of the letter used for the index is up to you, but must match with the letter used in the summand!

5. Properties of finite Series: This allows you to add the sums one of two ways. You can add the individual terms first and then sum all of them or you can sum the individual terms and add the two answers.

6. This allows you to either multiply each term by c then add the series, or first add the series and then multiply the result by c.

7. Example 1: Write each series in expanded form. a) The series starts with the first term and ends with the tenth term. It is summing 9k. Replace the numbers 1 through 10 in for k. 9 + 18 + 27 + 36 + 45 + 54 + 63 + 72 + 81 + 90

8. b) This series alternates in sign. Look at the (-1). It will alternate between positive and negative when k is odd or even. -2 + 4 - 6 + 8 - 10

9. c) Write in expanded form and then evaluate. Notice we used a different letter. Doesn't matter! 5 + 8 + 11 + 14 + 17 + 20 + 23 = 98

10. Example 2: Write each series using sigma notation. a)1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 This series is easy to spot. It's the sum of the squares from 1 to 9.

11. b) 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 Notice this is an arithmetic series. You must use the formula to find the explicit formula. d = 4, the first term is 5 and counting the terms, the eighth term is 33. t n = 5 + (n - 1)4 = 5 + 4n -4 = 4n +1

12. c) 1 - 1/3 + 1/9 - 1/27 + 1/81 - 1/243 Notice that this series is geometric. r = - 1/3, first term is 1 and the last term -1/243 is the 6 th term! Use the geometric formula! t n = 1( -1/3) (n - 1) = (-1/3) (n - 1)

13. HOMEWORK p. 508; 1 – 10 all