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Objectives

Learn the difference between finite and infinite series, write and evaluate arithmetic series, understand series summation, and interpret summation notation with practical examples.

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Objectives

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  1. Objectives 1. 2. Understand the difference between a finite and infinite series. Write and evaluate an arithmetic series.

  2. Understand the difference between Finite and Infinite Series Series: The sum of the terms of a sequence. Finite Sequences & Series: Can be counted and totaled (has an end). Infinite Sequences & Series: Cannot be counted and totaled (does not have an end). Finite Sequence 6, 9, 12, 15, 18 Finite Series 6 + 9 + 12 + 15 + 18 Infinite Sequence 3, 7, 11, 15, . . . Infinite Series 3 + 7 + 11 + 15 . . .

  3. Example 1: Writing & Evaluating a Series Use the finite sequence 2, 11, 20, 29, 38, 47. Write the related series, and then evaluate. Related Series: 2 + 11 + 20 + 29 + 38 + 47 = 147 Example 1: Practice • 21, 18, 15, 12, 9, 6, 3 2. 100, 99, 98, . . ., 95 3. 17.3, 19.6, 21.9, 24.2, 26.5 21 + 18 + 15 + 12 + 9 + 6 + 3; 84 100 + 99 + 98 + 97 + 96 + 95; 585 17.3 + 19.6 + 21.9 + 24.2 + 26.5; 109.5

  4. Arithmetic Series A series whose terms form an arithmetic sequence. When a sequence has many terms, or when you know only the 1st and last terms of the sequence, you can use a formula to evaluate the related series quickly. Sum of a Finite Arithmetic Series The sum Snof a finite arithmetic series a1 + a2 + a3 + . . . an is where a1is the 1st term, an is the nth term, & n is the # of terms.

  5. (5 + 13) 52 Example 2: Evaluating an Arithmetic Series Use the formula to evaluate the series related to the following sequence: 5, 7, 9, 11, 13 = = 2.5(18) = 45

  6. Example 2: Practice Each sequence has 8 terms. Evaluate each related series. 4. 32 5. 5, 13, 21, . . . , 61 6. 1,765, 1,414, 1,063, . . . , -692 264 4,292

  7. Objectives 1. 2. Discover the sum of the terms of an arithmetic series. Interpret summation notation and be able to rewrite as a series.

  8. 3  (5n + 1) n = 1 Using Summation Notation You can use the summation symbol  to write a series. Then you can use limits to indicate how many terms you are adding. Limits are the least and greatest integral value of n. upper limit, greatest value of n. explicit formula for the sequence lower limit, least value of n.

  9. The explicit formula for the sequence is ____. Example 3: Writing a Series in Summation Notation Use summation notation to write the series 3 + 6 + 9 + . . . for 33 items. 1 • 3 = 3 2 • 3 = 6 3 • 3 = 9 3n 33  The lower limit is ___. The upper limit is ___. 1 3 + 6 + 9 + . . . + 99 = 3n 33 1

  10. 4  2n n = 1 7  (n + 4) n = 1 15  7n n = 1 Example 3: Practice Use summation notation to write each arithmetic series for the specified number of terms. 13. 2 + 4 + 6 + . . . ; n = 4 15. 5 + 6 + 7 + . . . ; n = 7 17. 7 + 14 + 21 + . . . ; n = 15

  11. Example 4: Finding the Sum of a Series 3  (5n + 1). n = 1 3  (5n + 1) n = 1 To expand a series from summation notation, substitute each value of n into the explicit formula and add the results. Use the series a. Find the number of terms in the series. Since the values of n are 1, 2, & 3, there are ___ terms. 3 b. Find the first and last terms of the series. Term #1 = (5n + 1) = 5(1) + 1 = 6 Term #3 = (5n + 1) = 5(3) + 1 = 16 c. Evaluate the series. = = = 5(1) + 1 + 5(2) + 1 + 5(3) + 1 6 + 11 + 16 33

  12. 5  (2n - 1) n = 1 8  (7 - n) n = 3 4n3 10  n = 2 Example 4: Practice For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. 19. 21. 23. 5, 1, 9; 25 6, 4, -1; 9

  13. The price of the tickets 4 hours of fun 1 date 16 cups of fruit punch How many times Bobby tried to ask her but chickened out. Example 5: Real-World Application If Mary goes to the prom with Brian, he will be able to practice what he learned about sequences and series in Alg. 2 class. Questions 1. Is this series arithmetic, geometric, or neither? 1 + 4 + 16 + 64 + 256. geometric 2. Kayla, will you go to the prom with Bobby? ?

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