To evaluate an infinite geometric series means to find the sum of all the terms in that series, if that sum exists as a finite number. An infinite geometric series is the sum of the terms of an infinite geometric sequence, meaning it has no last term.
Understanding Infinite Geometric Series
An infinite geometric series takes the form:
a1 + a1r + a1r2 + a1r3 + ...
where:
- a1 is the first term
- r is the common ratio (the constant value multiplied by each term to get the next term)
Convergence and Divergence
Not all infinite geometric series can be evaluated. The ability to find a finite sum depends on the convergence of the series.
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Convergent Series: If the absolute value of the common ratio, |r|, is less than 1 (i.e., -1 < r < 1), the series converges, and its sum can be calculated. This is because each subsequent term becomes smaller and smaller, approaching zero.
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Divergent Series: If |r| is greater than or equal to 1 (i.e., r ≥ 1 or r ≤ -1), the series diverges. This means the terms either stay the same size or grow larger, and the sum approaches infinity (or negative infinity). Divergent series cannot be evaluated to a finite number.
Formula for the Sum of a Convergent Infinite Geometric Series
When an infinite geometric series converges (|r| < 1), the sum (S) can be calculated using the following formula:
S = a1 / (1 - r)
Examples
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Convergent Series: Consider the series 1 + 1/2 + 1/4 + 1/8 + ... Here, a1 = 1 and r = 1/2. Since |1/2| < 1, the series converges. The sum is S = 1 / (1 - 1/2) = 1 / (1/2) = 2.
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Divergent Series: Consider the series 1 + 2 + 4 + 8 + ... Here, a1 = 1 and r = 2. Since |2| ≥ 1, the series diverges. There is no finite sum for this series.
Summary
Evaluating an infinite geometric series means determining if the sum of its infinite terms approaches a finite value. This is only possible if the series converges, which occurs when the absolute value of the common ratio is less than 1. In this case, the sum can be calculated using the formula S = a1 / (1 - r). If the series diverges ( |r| ≥ 1), it cannot be evaluated to a finite sum.
