Infinite Geometric Series Objectives Find sums of infinite

Infinite Geometric Series

Objectives Find sums of infinite geometric series. Use mathematical induction to prove statements.

Vocabulary infinite geometric series converge limit diverge mathematical induction

In Lesson 12 -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.

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.

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.

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. B. 4 + 12 + 36 + 108 +. . . The series diverges and does not have a sum.

Try 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. The series converges and has a sum.

If an infinite series converges, we can find the sum. Consider the series from the previous page. Use the formula for the partial sum of a geometric series with and

Graph the simplified equation on a graphing calculator. Notice that the sum levels out and converges to 1. As n approaches infinity, the term approaches zero. Therefore, the sum of the series is 1. This concept can be generalized for all convergent geometric series and proved by using calculus.

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

Example 2 A Continued

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.

Example 2 B Continued

Try 2 a Find the sum of the infinite geometric series, if it exists. r = – 0. 2 Converges: |r| < 1. Sum formula = 125 6

Try 2 b Find the sum of the infinite geometric series, if it exists. Evaluate. Converges: |r| < 1

You can use infinite series to write a repeating decimal as a fraction.

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.

Example 3 Continued Step 2 Find the common ratio. |r | < 1; the series converges to a sum.

Example 3 Continued Step 3 Find the sum. Apply the sum formula. Check Use a calculator to divide the fraction

Remember! Recall that every repeating decimal, such as 0. 232323. . . , or 0. 23, is a rational number and can be written as a fraction.