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Calculator Answers

Infinite Series Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

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.

See also: Series Calculator, Taylor Series Calculator, and Maclaurin Series Calculator.

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 series

Infinite sum S

6

S = a / (1 − r) = 3 / (1 − 0.5) = 3 / 0.5 = 6

First term a3
Common ratio r0.5
|r|0.5
ConvergenceConverges (|r| < 1)
Preview (first 6 terms)3, 1.5, 0.75, 0.375, 0.1875, 0.09375

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.

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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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More tools from Education

Series CalculatorFind the nth term and the sum of the first n terms of an arithmetic or geometric sequence. Returns the closed-form formula used and a preview of the first several terms.Taylor Series CalculatorApproximate any differentiable function f(x) with a Taylor polynomial centered at a specified point a with successive derivatives.Maclaurin Series CalculatorCalculate the Maclaurin series expansion of any differentiable function f(x) centered at zero with worked step-by-step derivatives.Geometric Series CalculatorCalculate the nth term, partial sum, and infinite sum of a geometric series from the first term and common ratio.

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.