An infinite geometric series converges if and only if the absolute value of its common ratio is less than 1.
Here's a detailed explanation:
A geometric series is a series where each term is multiplied by a constant value, called the common ratio (r), to get the next term. An infinite geometric series continues indefinitely. Whether or not such a series adds up to a finite number (converges) depends on the value of 'r'.
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The Common Ratio (r): This is the constant factor between consecutive terms. For example, in the series 2 + 4 + 8 + 16 + ..., the common ratio is 2 (each term is multiplied by 2).
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Convergence Condition: An infinite geometric series converges (has a finite sum) if and only if
|r| < 1. This means -1 < r < 1. In simpler terms, the absolute value of the common ratio must be less than 1. -
Divergence Condition: If
|r| ≥ 1, the series diverges (does not have a finite sum). This means the terms either stay the same size or get larger and larger, so adding them up indefinitely will result in infinity (or negative infinity).
Why does this work?
When |r| < 1, each successive term in the series becomes smaller and smaller, approaching zero. The sum effectively "settles down" to a finite value as you add more and more terms.
When |r| ≥ 1, the terms either stay the same size (if r = 1 or r = -1) or grow larger in magnitude. Adding larger and larger terms will inevitably lead to a sum that grows without bound.
Examples:
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Convergent Series:
- Series: 1 + 1/2 + 1/4 + 1/8 + ...
- Common Ratio: r = 1/2
|r| = |1/2| = 1/2 < 1- Conclusion: This series converges. The sum is 2.
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Divergent Series:
- Series: 1 + 2 + 4 + 8 + ...
- Common Ratio: r = 2
|r| = |2| = 2 ≥ 1- Conclusion: This series diverges.
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Divergent Series (Oscillating):
- Series: 1 - 1 + 1 - 1 + 1 - ...
- Common Ratio: r = -1
|r| = |-1| = 1 ≥ 1- Conclusion: This series diverges. While it doesn't go to infinity, it oscillates between 0 and 1, not approaching a single finite value.
Summary:
To determine if an infinite geometric series converges, calculate the common ratio (r). If the absolute value of the common ratio is less than 1 (|r| < 1), the series converges. Otherwise, it diverges.
