Geometric Series Calculator

Calculate nth term, sum of n terms, and infinite limit of a geometric sequence.

Loading...

Features

Sequence Analysis

Calculate the nth number in the sequence and the sum of the first n terms.

Infinite Sums

Automatically calculates the sum to infinity if the series is convergent.

Convergence Check

Instantly identifies if your common ratio leads to a convergent or divergent series.

Sequence Preview

Visualizes the first few terms of the sequence to help you understand the pattern.

Formula Reference

Displays the mathematical formulas used for transparency and learning.

About Geometric Series Calculator

The Geometric Series Calculator is a powerful mathematical tool designed to analyze geometric progressions instantly. A geometric series is 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. This calculator handles both finite series (sum of first n terms) and infinite geometric series (convergent sum). It automatically detects if a series is convergent (|r| < 1) or divergent and provides the appropriate sums.

How to Use Geometric Series Calculator

  • 1
    Enter First Term

    Input the starting value (a) of your geometric sequence.

  • 2
    Set Common Ratio

    Input the ratio (r) by which each term is multiplied.

  • 3
    Specify Terms

    Optional: Enter the number of terms (n) to calculate a specific partial sum.

  • 4
    Analyze Results

    View the nth term, partial sum, and infinite sum (if applicable).

Frequently Asked Questions

It is 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 (r).
A 'sequence' is simply an ordered list of numbers. A 'series' is the mathematical sum of the terms in that sequence.
A geometric series converges (approaches a finite limit) only if the absolute value of the common ratio is less than 1 (|r| < 1).
If the series converges, the sum to infinity (S∞) is calculated as: S∞ = a / (1 - r), where 'a' is the first term.
Yes, a negative ratio causes the terms to alternate between positive and negative values. Convergence still depends on the absolute value being less than 1.
If r = 1, every term is identical to the first term, and the sum of 'n' terms is simply 'n × a'. The series does not converge to an infinite sum.