Sum of Geometric Series
Sₙ = a₁(1 − rⁿ) / (1 − r) for r ≠ 1. Infinite: S∞ = a₁/(1−r) if |r|<1.
Result
S∞
2.000000
Infinite geometric series sum.
- a₁1
- r0.5
- ninfinite
- Sum2.000000
Step-by-step
- S∞ = a₁ / (1 − r) = 1 / 0.5 = 2.0000.
How to use this calculator
- Enter a₁, r, and n.
- Use n=0 for infinite series (only valid |r|<1).
About this calculator
Sum of finite geometric series: Sₙ = a₁(1 − rⁿ)/(1 − r). Infinite series only converges if |r| < 1: S∞ = a₁/(1−r). Examples: Zeno's paradox (1 + ½ + ¼ + ... = 2), present value of perpetuity in finance (CF/r), exponential decay totals. The formula is a cornerstone of finance, probability, and physics (geometric distribution).
Frequently asked
Take limit n → ∞. r^n → 0 if |r|<1. Sₙ → a₁/(1−r).
Related calculators
Geometric Sequence Calculator
aₙ = a₁ × r^(n−1). Find any term of a geometric sequence.
Sum of Arithmetic Series
Sₙ = n × (a₁ + aₙ) / 2 = n/2 × (2a₁ + (n−1)d).
Arithmetic Sequence Calculator
aₙ = a₁ + (n−1)d. Find any term of an arithmetic sequence.
Fibonacci Number Calculator
F(n) = F(n−1) + F(n−2). 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Harmonic Mean Calculator
H = n / Σ(1/xᵢ). Used for averaging rates and ratios.
Weighted Average Calculator
WA = Σ(value × weight) / Σ(weight). Generalizes the arithmetic mean.