Geometric Series Objectives 1. 2. Evaluate a finite and infinite geometric series. Understand the difference between convergence and divergence, and identify.

Evaluating a Finite Geometric Series Sum of a Finite Geometric Series The sum Sn of a finite geometric series a1 + a2 + a3 + . . . an is where a1 is the 1st term, r is the common ratio, & n is the # of terms.

Example 1: Using the Geometric Series Formula Use the formula to evaluate the series 3 + 6 + 12 + 24 + 48 + 96. The first term is ___, and there are ___ terms in the series. 3 6 The common ratio is = 2 So a1 = ___, r = ___, and n = ___. 3 2 6 189

Example 2: Real-World Connection Financial Planning: In March, Larry, Moe and Curly start saving for a vacation at the end of August. The Stooges expect the vacation to cost $1,375. They start with $125. Each month they plan to deposit 20% more than the previous month. Will they have enough $ for their trip? Write the geometric series formula. Sn = Total amount to be saved. a1 = $125 r = 1.2 n = 6 $1,243.75

Examples 1 & 2: Practice 1. 1 + 2 + 4 + . . . ; n = 8 Evaluate the finite series for the specified number of terms. 1. 1 + 2 + 4 + . . . ; n = 8 2. 3 + 6 + 12 + . . . ; n = 7 3. -5 – 10 – 20 – . . . ; n = 11 255 381 -10,235 255256 4.

Infinite Geometric Series: Convergence vs. Divergence In some cases, you can evaluate an infinite geometric series. When │r│< 1, the series converges (i.e. gets closer and closer to the sum S). When │r│≥ 1, the series diverges (i.e. it approaches no limit).

 Example 3: Determining Divergence & Convergence A. B. a1 = 1 a1 = 5 Decide whether each infinite geometric series diverges or converges. State whether the series has a sum.  5(2)n-1 ∞ n = 1 A. B. a1 = 1 a1 = 5 a2 = -⅓ a2 = 10 r = -⅓  1 = -⅓ r = 10  5 = 2 Since │r │ < 1, the series converges & has a sum. Since │r │ > 1, the series diverges & does not have a sum.

Example 3: Practice Decide whether each infinite geometric series diverges or converges. State whether the series has a sum. 5. 6. 7. 8. 9. converges; has a sum diverges; no sum 4 + 2 + 1 . . . 6 + 18 + 54 + . . . 1 - 1 + 1 - . . . ¼ + ½ + 1 + 2 . . .

Sum of an Infinite Geometric Series An infinite geometric series with │r│ < 1 converges to the sum of where a1 is the 1st term, r is the common ratio. You can use the sum formula to evaluate some infinite geometric series.

Example 4: Real-World Connection The length of the outside shell of each closed chamber of a chambered nautilus is 0.9 times the length of the larger chamber next to it. What is the best estimate for the total length of the outside shell for the enclosed chambers? 27mm S1 = 27 = 270 mm

Example 4: Practice Evaluate each infinite geometric series. 10. 11. 12. 1.1 – 0.11 + 0.011 - . . . 1 92 95