Infinite Series Calculator
Calculate the sum of an infinite geometric series given the first term and common ratio.
Published by:
isCalculator Editorial Team
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How to Use the Infinite Geometric Series Calculator
This calculator finds the sum of an infinite geometric series using the formula S∞ = a / (1 - r), where a is the first term and r is the common ratio.
- Enter the first term (a) in the "First Term" field. Use any real number (e.g., 1, -3, 0.5).
- Enter the common ratio (r) in the "Common Ratio" field. The series only converges if |r| < 1 (e.g., 0.5, -0.25, 0.99).
- Click "Calculate Sum" or press Enter on your keyboard.
- The result shows:
- The sum (S∞) rounded to 6 decimal places.
- The formula explanation with your numbers substituted.
- If you see an error, check that:
- Both fields contain valid numbers.
- The common ratio satisfies |r| < 1 (otherwise the series diverges).
Example: a = 1, r = 0.5 → Sum = 1 / (1 - 0.5) = 2.000000
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Frequently Asked Questions
An infinite geometric series is the sum of an infinite number of terms that follow a geometric progression, where each term is obtained by multiplying the previous term by a constant ratio.
An infinite geometric series converges if the absolute value of the common ratio is less than 1 (|r| < 1). The sum is given by a/(1 - r), where a is the first term.
If |r| ≥ 1, the series diverges and does not have a finite sum. The calculator will show an error message.
