Algebra 101 - Unit 11

Lesson 69: Infinite Geometric Series

An infinite geometric series is a geometric sequence whose terms continue indefinitely. The sum of an infinite geometric series exists only if the common ratio r satisfies |r| < 1.

Sum Formula

For an infinite geometric series with first term a₁ and common ratio r (|r| < 1): S∞ = a₁ / (1 − r)

Examples

Example 1: Find the sum of 3, 1.5, 0.75, …

Solution: r = 0.5 → S∞ = 3 / (1 − 0.5) = 3 / 0.5 = 6

Example 2: Find the sum of 8, −4, 2, −1, …

Solution: r = −0.5 → S∞ = 8 / (1 − (−0.5)) = 8 / 1.5 ≈ 5.333

Example 3: Determine if the series 5, 10, 20, … has a sum.

Solution: r = 2, |r| > 1 → The series diverges, so no sum exists.

Practice Problems

1. Find the sum of the infinite series 1, 0.3, 0.09, …
2. Determine the sum of 12, −6, 3, −1.5, …
3. Does the series 7, 14, 28, … have a sum? Explain.
4. Find S∞ for the series 0.5, 0.25, 0.125, …
5. Find the sum of the infinite geometric series −2, 1, −0.5, 0.25, …