Soma PG Infinita
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
S = a/(1−r)
About this calculator
The infinite geometric series calculator determines the sum of all terms of a never-ending geometric progression, provided the ratio is between -1 and 1. In this case, the terms become increasingly smaller and the sum converges to a finite number. The formula used is S = a/(1 - r), where 'a' is the first term and 'r' is the ratio. The calculator automatically checks if |r| < 1; otherwise, it informs that the sum does not converge.
To use the tool, enter the first term and the ratio. The result is displayed immediately. For example, if a = 1 and r = 0.5, the sum will be 1/(1 - 0.5) = 2. This means the infinite sum 1 + 0.5 + 0.25 + ... equals 2. The calculator is useful for math, physics, and finance problems involving infinite series.
Important precautions: the formula only works if |r| < 1. If the ratio is greater than or equal to 1 in absolute value, the series diverges and has no finite sum. Also, ensure the first term is not zero, as that would make the sum zero. The calculator handles decimal and fractional numbers, but avoid very large inputs to prevent rounding errors.
Frequently asked questions
What does |r| < 1 mean?
It means the ratio must be between -1 and 1, excluding the endpoints. For example, r = 0.5 or r = -0.3 are valid, but r = 1 or r = -2 are not.
What if the ratio equals 1?
If r = 1, the series is constant and the infinite sum diverges to infinity (or negative infinity if a is negative). The calculator will indicate that the sum does not converge.
Can I use fractional numbers?
Yes, you can enter fractions like 1/2 or 3/4, but it is recommended to use the decimal equivalent (0.5, 0.75) for better accuracy.
Does the calculator work for negative ratios?
Yes, it works. For example, a = 2 and r = -0.5 gives S = 2/(1 - (-0.5)) = 2/1.5 ≈ 1.333. The series alternates signs but converges.