How Do You Find the Infinite Geometric Sum?
You can find the sum of an infinite geometric series if its common ratio (r) is between -1 and 1 (i.e., |r| < 1). Otherwise, the sum diverges (goes to infinity).
An infinite geometric series is a sum of infinitely many terms forming a geometric sequence. A geometric sequence is a series where each term is the previous term multiplied by a constant value called the common ratio (r). For example, 1 + 1/2 + 1/4 + 1/8 + ... is an infinite geometric series with a common ratio of 1/2. Learn more about geometric series.
Formula for the Sum
The sum (S) of an infinite geometric series is calculated using the formula:
S = a / (1 - r)
Where:
- a is the first term of the series.
- r is the common ratio. This value MUST be between -1 and 1 for the sum to converge to a finite value. As noted in several sources, if |r| ≥ 1, the sum diverges and doesn't have a finite value. (See Reddit discussion, See Math StackExchange discussion).
Examples
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Example 1: Find the sum of the series 1 + 1/2 + 1/4 + 1/8 + ...
Here, a = 1 and r = 1/2. Since |r| < 1, the sum converges:
S = 1 / (1 - 1/2) = 2
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Example 2: Find the sum of the series 36 + 12 + 4 + 4/3 + ...
Here, a = 36 and r = 1/3. Since |r| < 1, the sum converges:
S = 36 / (1 - 1/3) = 54
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Example 3: The series 1 + 2 + 4 + 8 + ... diverges because r = 2, and |r| > 1. (See Mathway example for finite series)
Practical Insights
The formula for the sum of an infinite geometric series has applications in various fields, including:
- Finance: Calculating the present value of a perpetuity (an annuity that pays indefinitely).
- Physics: Modeling phenomena involving exponential decay.
- Computer science: Analyzing algorithms with recursive structures.
