# Geometric Series Sum Calculator — nth Term & S∞

> Geometric series calculator: find the nth term a·rⁿ⁻¹, the partial sum Sₙ, and the infinite sum a/(1−r) when |r|<1. Shows the working and convergence.

- **Category:** Math
- **Interactive calculator:** https://youcalc.com/en/math/geometric-series-sum/
- **Price:** Free, no sign-up required

## Overview

This calculator works with a geometric series — a sequence where each term is the previous one multiplied by a fixed common ratio r, starting from a first term a. Enter a, r and the number of terms n to instantly see the nth term aₙ, the partial sum of the first n terms Sₙ, and the sum to infinity S∞ when the series converges.

## How to read your result

The result card shows the partial sum Sₙ of the first n terms at the top. Below it you will find the nth (last) term aₙ = a·rⁿ⁻¹ and the sum to infinity. The sum to infinity is a finite number only when |r| < 1, because that is the condition for the terms to shrink toward zero fast enough for the total to settle; if |r| ≥ 1 the series diverges and the calculator says so instead of printing a misleading number. The worked solution underneath substitutes your values into each closed form so you can follow every step.

## Method

The nth term is computed with the closed form aₙ = a·rⁿ⁻¹. The partial sum uses Sₙ = a·(1 − rⁿ)/(1 − r) for r ≠ 1, falling back to Sₙ = n·a when r = 1 to avoid dividing by zero. The sum to infinity is S∞ = a/(1 − r) and is reported only when |r| < 1, the standard convergence condition for a geometric series; otherwise the calculator states that the series diverges rather than printing a value. These formulas and the convergence rule are documented by Wolfram MathWorld and Wikipedia.

## Example

- **Setup:** Enter a = 1, r = 2, n = 10 (the series 1 + 2 + 4 + … + 512).
- **Result:** The nth term is a₁₀ = 1·2⁹ = 512 and the partial sum is S₁₀ = 1·(1 − 2¹⁰)/(1 − 2) = 1023. Because |r| = 2 ≥ 1, the series diverges, so there is no sum to infinity. Switch to r = 0.5 and the sum to infinity becomes a/(1 − r) = 1/0.5 = 2.

## Frequently asked questions

### When does a geometric series have a sum to infinity?

A geometric series converges to a finite sum to infinity only when the common ratio satisfies |r| < 1. In that case S∞ = a/(1 − r). When |r| ≥ 1 — including r = 1 and r = −1 — the terms do not shrink toward zero, the partial sums keep growing or oscillating, and there is no finite total, so the series is said to diverge.

### What is the difference between the nth term and the partial sum?

The nth term aₙ = a·rⁿ⁻¹ is the value of a single term — the nth entry in the sequence. The partial sum Sₙ adds up the first n terms together: Sₙ = a·(1 − rⁿ)/(1 − r) for r ≠ 1, or simply n·a when r = 1. So aₙ is one number from the list, while Sₙ is the running total of the whole list up to that point.

### Can the first term or common ratio be negative?

Yes. The first term a and the common ratio r can be any finite numbers, including negatives and fractions. A negative ratio makes the terms alternate in sign (for example 1, −2, 4, −8…). The closed-form formulas handle every case; the only requirement for a finite sum to infinity is still |r| < 1.

### Why does the formula change when r = 1?

The general partial-sum formula Sₙ = a·(1 − rⁿ)/(1 − r) divides by (1 − r), which is zero when r = 1. In that case every term equals a, so the sum is simply n copies of a: Sₙ = n·a. The calculator detects r = 1 and uses this special case automatically.

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## Sources

- https://mathworld.wolfram.com/GeometricSeries.html
- https://en.wikipedia.org/wiki/Geometric_series

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