Mathematical Induction and 9 5 Infinite Geometric Series

Mathematical Induction and 9 -5 Infinite Geometric Series Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Algebra 2 Holt

Mathematical Induction and 9 -5 Infinite Geometric Series Warm Up Evaluate. 1. 2. 10 3. Write 0. 6 as a fraction in simplest form. 4. Find the indicated sum for the geometric series Holt Mc. Dougal Algebra 2 3

Mathematical Induction and 9 -5 Infinite Geometric Series Objectives Find sums of infinite geometric series. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Essential Question How do you determine whether a geometric series converges or diverges? Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series In Lesson 9 -4, you found partial sums of geometric series. You can also find the sums of some infinite geometric series. An infinite geometric series has infinitely many terms. Consider the two infinite geometric series below. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Notice that the series Sn has a common ratio of and the partial sums get closer and closer to 1 as n increases. When |r|< 1 and the partial sum approaches a fixed number, the series is said to converge. The number that the partial sums approach, as n increases, is called a limit. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series For the series Rn, the opposite applies. Its common ratio is 2, and its partial sums increase toward infinity. When |r| ≥ 1 and the partial sum does not approach a fixed number, the series is said to diverge. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 1: Finding Convergent or Divergent Series Determine whether each geometric series converges or diverges. A. 10 + 1 + 0. 01 +. . . The series converges and has a sum. Holt Mc. Dougal Algebra 2 B. 4 + 12 + 36 + 108 +. . . The series diverges and does not have a sum.

Mathematical Induction and 9 -5 Infinite Geometric Series Check It Out! Example 1 Determine whether each geometric series converges or diverges. A. B. 32 + 16 + 8 + 4 + 2 + … The series diverges and does not have a sum. Holt Mc. Dougal Algebra 2 The series converges and has a sum.

Mathematical Induction and 9 -5 Infinite Geometric Series If an infinite series converges, we can find the sum using a simpler formula than the one used for partial sums. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 2 A: Find the Sums of Infinite Geometric Series Find the sum of the infinite geometric series, if it exists. 1 – 0. 2 + 0. 04 – 0. 008 +. . . r = – 0. 2 Converges: |r| < 1. Sum formula Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 2 A Continued Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 2 B: Find the Sums of Infinite Geometric Series Find the sum of the infinite geometric series, if it exists. Evaluate. Converges: |r| < 1. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 2 B Continued Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Check It Out! Example 2 a Find the sum of the infinite geometric series, if it exists. r = – 0. 2 Converges: |r| < 1. Sum formula 125 = 6 Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Check It Out! Example 2 b Find the sum of the infinite geometric series, if it exists. Evaluate. Converges: |r| < 1 Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series You can use infinite series to write a repeating decimal as a fraction. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 3: Writing Repeating Decimals as Fractions Write 0. 63 as a fraction in simplest form. Step 1 Write the repeating decimal as an infinite geometric series. 0. 636363. . . = 0. 63 + 0. 000063 +. . . Use the pattern for the series. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 3 Continued Step 2 Find the common ratio. |r | < 1; the series converges to a sum. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Example 3 Continued Step 3 Find the sum. Apply the sum formula. Check Use a calculator to divide the fraction Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Remember! Recall that every repeating decimal, such as 0. 232323. . . , or 0. 23, is a rational number and can be written as a fraction. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Check It Out! Example 3 Write 0. 111… as a fraction in simplest form. Step 1 Write the repeating decimal as an infinite geometric series. Use the pattern 0. 111. . . = 0. 1 + 0. 001 +. . . for the series. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Check It Out! Example 3 Continued Step 2 Find the common ratio. |r | < 1; the series converges to a sum. Step 3 Find the sum. Apply the sum formula. Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Lesson Quiz Solve each equation. 1. Determine whether the geometric series 150 + 30 + 6 + … converges or diverges, and find the sum if it exists. converges; 187. 5 4 2. Write 0. 0044 as a fraction in simplest form. 909 Holt Mc. Dougal Algebra 2

Mathematical Induction and 9 -5 Infinite Geometric Series Essential Question How do you determine whether a geometric series converges or diverges? When |r|< 1 and the partial sum approaches a fixed number, the series is said to converge. When |r| ≥ 1 and the partial sum does not approach a fixed number, the series is said to diverge. Holt Mc. Dougal Algebra 2