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Geometric Series Calculator

Last updated: June 17, 2026

Written by Blake Boege · Founder, Calculator Answers

A geometric series calculator analyzes a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The calculator computes the value of the nth term, the sum of the first n terms (partial sum), and the sum of the infinite series if the absolute value of the common ratio is less than one. If the ratio is greater than or equal to one, the calculator indicates that the infinite series diverges.

Find the nth term, finite partial sum, and infinite sum of a geometric sequence.

Quick Answer

Analyze a geometric progression. Enter the first term, common ratio, and number of terms to find the nth term, partial sum, and infinite sum.

First term (a)

The starting term of the series. · e.g. 2

Common ratio (r)

The multiplier between consecutive terms. · e.g. 0.5

Number of terms (n)

A positive integer representing the term count. · e.g. 5

Geometric Series Results

Partial Sum S_n (n = 5)

3.875

nth Term = 0.125

First term (a)2

Common ratio (r)0.5

Number of terms (n)5

nth term (a_n)0.125

Partial Sum (S_n)3.875

Infinite Sum (S_inf)4

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Examples

Converging Series

first term = 2 · ratio = 0.5 · terms = 5 — nth term = 0.125 · S_n = 3.875 · Infinite Sum = 4.000

Diverging Series

first term = 3 · ratio = 2.0 · terms = 4 — nth term = 24 · S_n = 45 · Infinite Sum = Diverges

How it works

Formula Definitions

For a geometric series with first term a and common ratio r:

nth term (a_n): a_n = a × r^(n-1)
Partial Sum (S_n):
If r ≠ 1: S_n = a × (1 - r^n) / (1 - r)
If r = 1: S_n = a × n
Infinite Sum (S_inf):
If |r| < 1: S_inf = a / (1 - r) (converges)
If |r| ≥ 1: The series diverges.

Disclaimer

For educational purposes. Verify results for critical applications.

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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.Infinite Series CalculatorTest an infinite geometric series for convergence and find the sum when |r| < 1 using S = a / (1 − r). Returns the sum, the convergence verdict, and a preview of the first few terms.Radius of Convergence CalculatorCalculate the radius of convergence R of any power series using the Ratio Test with numerical limit approximations.

Frequently asked questions

A geometric series is the sum of a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed constant called the common ratio (r).

For a finite series of n terms, the partial sum is calculated using S_n = a × (1 - r^n) / (1 - r), provided the common ratio r is not equal to 1. If r = 1, the sum is simply a × n.

An infinite geometric series converges if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). If |r| ≥ 1, the series diverges (approaches infinity or oscillates infinitely).

If the series converges (|r| < 1), the sum of the infinite series is calculated using the formula: S_inf = a / (1 - r).

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