The convergence of a geometric series depends entirely on the value of its common ratio, r.
Understanding Geometric Series Convergence
A geometric series is a series with a constant ratio between successive terms. It takes the form:
a + ar + ar2 + ar3 + ...
where:
- a is the first term
- r is the common ratio
The critical factor determining convergence is the absolute value of the common ratio, |r|.
Conditions for Convergence and Divergence
Here's a breakdown of when a geometric series converges or diverges:
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Convergence: An infinite geometric series converges if and only if the absolute value of the common ratio is less than 1: | r | < 1. In this case, the sum converges to a finite value, which can be calculated using the formula:
S = a / (1 - r)
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Divergence: An infinite geometric series diverges if the absolute value of the common ratio is greater than or equal to 1: | r | ≥ 1. In this situation, the sum of the series approaches infinity (or negative infinity) and does not converge to a finite value.
Finite Geometric Series: A finite geometric series always converges, regardless of the value of r. This is because you are only summing a finite number of terms, so the sum will always be a finite number.
Examples
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Convergent Series: 1 + 1/2 + 1/4 + 1/8 + ... Here, a = 1 and r = 1/2. Since |1/2| < 1, the series converges. The sum is 1 / (1 - 1/2) = 2.
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Divergent Series: 1 + 2 + 4 + 8 + ... Here, a = 1 and r = 2. Since |2| > 1, the series diverges.
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Divergent Series: 1 - 1 + 1 - 1 + ... Here, a = 1 and r = -1. Since |-1| = 1, the series diverges.
Summary
An infinite geometric series converges only when the absolute value of its common ratio is less than 1 (|r| < 1). Otherwise, it diverges. Finite geometric series always converge.
