Education
Infinite Series Calculator
Enter the first term a and the common ratio r. The calculator checks |r| against 1, reports convergence or divergence, and computes the infinite sum when convergent.
Infinite geometric series
a + a·r + a·r² + a·r³ + … Converges when the absolute value of the common ratio r is strictly less than 1.
Convergence requires |r| < 1.
Scope
Infinite geometric series only. For arithmetic and geometric finite sums (nth term and sum of the first n terms), use the series calculator.
Out of scope: p-series, alternating series convergence tests, ratio/root tests for arbitrary series, and divergent series summation methods.
Infinite sum S
6
S = a / (1 − r) = 3 / (1 − 0.5) = 3 / 0.5 = 6
The sum S = a / (1 − r) is the limit of the partial sums as n grows. If |r| ≥ 1, the partial sums do not settle to a finite number and the series diverges.
Examples
a = 3, r = 0.5
Converges; S = 3 / 0.5 = 6
a = 1, r = 0.9
Converges; S = 1 / 0.1 = 10
a = 1, r = 2
Diverges (|r| ≥ 1)
a = 5, r = −0.25
Converges; S = 5 / 1.25 = 4
How it works
The infinite geometric series sum comes from the closed-form formula for the finite sum Sₙ = a · (1 − rⁿ) / (1 − r). When |r| < 1, rⁿ → 0 as n → ∞, leaving S = a / (1 − r).
Sum (converges) · S = a / (1 − r) when |r| < 1
Diverges · |r| ≥ 1
Related calculators
- Series calculator for arithmetic and geometric nth term and finite sum of the first n terms.
- Exponent calculator for the rⁿ side of the finite-sum formula.
- Fraction calculator when a or r is a fraction.
- Scientific calculator for general arithmetic checks.
- All education calculators.
Frequently asked questions
A sum a + a·r + a·r² + a·r³ + … with first term a and constant common ratio r. Each term is the previous term multiplied by r.
When |r| < 1. In that case the terms shrink quickly enough that the partial sums approach a finite limit. The limit is S = a / (1 − r).
When |r| > 1, terms grow without bound. When |r| = 1, terms stay at constant magnitude (either equal to a or alternating). In both cases the partial sums do not settle to a finite value.
Every term is zero and the sum is trivially 0. The calculator handles this as a converging case with sum 0.
The series calculator handles arithmetic and geometric sequences and computes finite sums (nth term and sum of the first n terms). This page is specifically for the infinite sum of a geometric series when it converges.
No. Only geometric series. For p-series, alternating series, ratio test, or root test, work them through the standard convergence tests by hand or use a CAS.
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