Common Ratio
The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. The following table shows several geometric series with different common ratios:
| Common ratio | Example |
|---|---|
| 10 | 4 + 40 + 400 + 4000 + 40,000 + ··· |
| 1/3 | 9 + 3 + 1 + 1/3 + 1/9 + ··· |
| 1/10 | 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ··· |
| 1 | 3 + 3 + 3 + 3 + 3 + ··· |
| −1/2 | 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ··· |
| –1 | 3 − 3 + 3 − 3 + 3 − ··· |
The behavior of the terms depends on the common ratio r:
- If r is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in the limit and the series converges to a sum. In the case above, where r is one half, the series has the sum one.
- If r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series diverges.)
- If r is equal to one, all of the terms of the series are the same. The series diverges.
- If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum. See for example Grandi's series: 1 − 1 + 1 − 1 + ···.
Read more about this topic: Geometric Series
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