Série geométrica infinita
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The infinite geometric series calculator computes the sum of a series in the form a + a*r + a*r² + a*r³ + ..., provided the common ratio r is between -1 and 1 (|r| < 1). This series converges to a finite value given by the formula S = a / (1 - r), where 'a' is the first term and 'r' is the ratio. It is widely used in mathematics, engineering, and sciences to model situations involving infinite sums with consistent term reduction.
To use the calculator, input the first term (a) and the ratio (r). The program automatically checks if |r| < 1 and, if valid, applies the formula. If the ratio exceeds this limit, the series diverges and the result becomes invalid. Convergence only occurs when each term is a fraction of the previous one, ensuring the total sum stabilizes.
This tool is useful in compound interest calculations, radioactive decay modeling, and any context where proportional term reduction is constant. Remember the formula only works for infinite series; use a different calculation for finite series. Always verify the ratio falls within the valid range before applying the formula.
Frequently asked questions
What formula is used to calculate the infinite geometric series?
The formula is S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. It only works when |r| < 1.
Why must the ratio be between -1 and 1?
If |r| ≥ 1, terms don't shrink, and the series diverges (no finite sum). Convergence is guaranteed only when |r| < 1.
Can I use this calculator for finite series?
No. For a finite number of terms, use the finite geometric series formula: S = a*(1 - r^n)/(1 - r).
What happens if the ratio is exactly 1?
If r = 1, the series becomes a + a + a + ..., with an infinite sum. The calculator won't provide a valid result in this case.
Are the results 100% accurate?
Accuracy depends on the calculator, but calculations follow the exact mathematical formula. Limitations occur only with input errors.