What is a geometric series?

A geometric series is the sum of terms in a geometric sequence, where each term is obtained by multiplying the previous term by a fixed number called the common ratio. For example: 2 + 6 + 18 + 54 (ratio = 3).

When does an infinite geometric series converge?

An infinite geometric series converges (has a finite sum) only when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges (sum approaches infinity).

What if the common ratio is 1?

When r = 1, all terms are equal to the first term. The sum is simply n × a, where n is the number of terms and a is the first term.

Can the common ratio be negative?

Yes! A negative common ratio creates an alternating series. For example, with a = 2 and r = -3: 2, -6, 18, -54, 162. The calculator handles negative ratios correctly.

Geometric Series Sum Calculator

Calculate the sum of geometric sequences instantly. Find the sum of finite or infinite geometric series with step-by-step solutions. Perfect for students and math professionals.

First Term (a)

Common Ratio (r)

Number of Terms (n)

Enter values to calculate geometric series sum

Geometric Series Sum Calculator Tool Introduction

A comprehensive Geometric Series Sum Calculator that computes the sum of geometric sequences with ease. Enter the first term, common ratio, and number of terms to get instant results with detailed explanations.

Perfect for students studying sequences and series, teachers preparing lessons, engineers working with exponential growth, and anyone dealing with geometric progressions in finance, physics, or computer science.

Our calculator handles both finite and infinite geometric series, showing you the formula used and step-by-step calculations. It also indicates when an infinite series converges or diverges based on the common ratio.

All calculations are performed locally in your browser for complete privacy. No data is collected or stored. Use this tool to verify homework, understand geometric series, or solve real-world problems involving exponential patterns.

Geometric Series Sum Calculator User Guide

How to Use

Enter First Term (a): The initial value of the sequence
Enter Common Ratio (r): The multiplier between consecutive terms
Enter Number of Terms (n): How many terms to sum
Click Calculate: Get the sum and detailed explanation

Understanding Geometric Series

A geometric series is the sum of terms in a geometric sequence, where each term is found by multiplying the previous term by a constant called the common ratio.

Formula

Finite Series: S_n = a(1 - rⁿ) / (1 - r) when r ≠ 1

When r = 1: S_n = n × a

Infinite Series (|r| < 1): S = a / (1 - r)

Example

Sum of 2 + 6 + 18 + 54 + 162:

First term (a) = 2
Common ratio (r) = 3
Number of terms (n) = 5
Sum = 2(1 - 3⁵) / (1 - 3) = 2(-242) / (-2) = 242

Key Features

Calculate finite geometric series sums
Evaluate infinite series convergence
Step-by-step solution display
Support for negative ratios
Handles fractional ratios
Shows all terms in the sequence

Frequently Asked Questions

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