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Infinite Series Calculator
Last updated: June 19, 2026
An infinite series calculator is an advanced mathematics tool that evaluates the behavior of a sequence summed to infinity. The calculator analyzes the terms of the series and applies standard calculus tests, including the geometric series test, ratio test, integral test, and p-series test, to determine if the summation converges to a finite value or diverges. For convergent geometric or telescoping series, it computes the exact sum to infinity. Calculus students and mathematicians use this tool to verify series convergence behavior.
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.
Quick Answer
Calculate the sum and convergence of infinite series. Enter the formula and variable parameters to determine convergence limits and values.
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
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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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