Infinite Geometric Series
A geometric series with infinitely many terms, a + ar + ar² + …,
converges when |r| < 1, and its sum is S = a/(1 − r)
diverges when |r| ≧ 1 (excluding a = 0)
Example ① Find the sum of 1 + 1/2 + 1/4 + 1/8 + …
Solution: Since |r|=1/2 < 1, the series converges.
S = 1/(1−1/2) = 2
Solution: Since |r|=1/2 < 1, the series converges.
S = 1/(1−1/2) = 2
Example ② Express the repeating decimal 0.333… as a fraction.
Solution:0.3 + 0.03 + 0.003 + … = 0.3/(1−0.1) = 0.3/0.9 = 1/3
Solution:0.3 + 0.03 + 0.003 + … = 0.3/(1−0.1) = 0.3/0.9 = 1/3
Key Point
- |r| < 1 is an absolute condition
- It can be applied to express repeating decimals as rational numbers
Practice Problems
- Find the sum of 2 + 2/3 + 2/9 + …
Answer and Explanation
- Answer:3
2/(1−1/3)=2·3/2=3.