Infinite Series Calculator — Sum of Geometric Series & Convergence Checker

Calculate the sum of an infinite geometric series instantly. Check convergence conditions, get step-by-step formula breakdowns, and compute infinite series sums online with our free calculator.

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Infinite Series Calculator

Enter the first term and common ratio to find the sum of an infinite geometric series, or enter the first two terms to auto-detect the ratio.

Enter values and click Calculate Sum to see the result.

Infinite Geometric Series Formula Explained

The sum of an infinite geometric series is given by the formula below. This formula applies when the absolute value of the common ratio is less than 1, ensuring the series converges to a finite value.

S∞ = a / (1 − r)    where   |r| < 1

Variable Definitions

  • a (or a₁) — The first term of the infinite geometric series
  • r — The common ratio between consecutive terms (r = a₂/a₁ = a₃/a₂)
  • S∞ — The sum of all infinitely many terms, which exists only if |r| < 1

If |r| ≥ 1, the series diverges and does not have a finite sum. The terms either grow without bound or oscillate indefinitely.

How to Calculate the Sum of an Infinite Geometric Series

Follow these steps to calculate the sum of any convergent infinite geometric series:

  1. Identify the first term a — This is the starting value of your series (e.g., 72 in the series 72 + 60 + 50 + …).
  2. Find the common ratio r — Divide any term by the previous term: r = a₂/a₁. For the example above, r = 60/72 = 5/6 ≈ 0.8333.
  3. Check convergence — Verify that |r| < 1. If true, the series converges to a finite sum. If |r| ≥ 1, the series diverges.
  4. Apply the formula — Use S∞ = a / (1 − r).
  5. Simplify the result — Compute the final value. For 72/(1−5/6) = 72/(1/6) = 432.

Infinite Series Calculator Examples

Example 1: Classic Geometric Series

Find the sum of the infinite series 72 + 60 + 50 + …

a = 72,   r = 60/72 = 5/6 ≈ 0.8333
|r| = 0.8333 < 1 → Converges
S∞ = 72 / (1 − 5/6) = 72 / (1/6) = 432

Example 2: Series with Negative Ratio

Find the sum of 20 − 10 + 5 − 2.5 + …

a = 20,   r = −10/20 = −0.5
|r| = 0.5 < 1 → Converges
S∞ = 20 / (1 − (−0.5)) = 20 / 1.5 ≈ 13.3333

Example 3: Divergent Series

Check the series 3 + 6 + 12 + 24 + …

a = 3,   r = 6/3 = 2
|r| = 2 ≥ 1 → Diverges (no finite sum)

Real-World Infinite Series Applications

  • Calculating Pi: Infinite series like the Gregory-Leibniz series (π/4 = 1 − 1/3 + 1/5 − 1/7 + …) approximate π to high precision.
  • Physics & Motion: Zeno's paradox and the sum of an infinite geometric series describe motion where an object covers half the remaining distance repeatedly.
  • Finance & Economics: Perpetuity formulas use infinite geometric series to value assets that pay indefinitely (e.g., preferred stock dividends).
  • Fractals & Geometry: The area and perimeter of fractal shapes like the Koch snowflake are computed using infinite series.
  • Signal Processing: Fourier series represent periodic signals as infinite sums of sine and cosine waves for audio, image, and data compression.
  • Probability: Infinite geometric series calculate expected values in repeated trials, such as the expected number of coin flips until heads appears.
  • Computer Science: Taylor series expansions approximate functions like ex, sin(x), and cos(x) for numerical computation in software.

People Also Ask About Infinite Series

The sum is S∞ = a/(1−r), where a is the first term and r is the common ratio. This formula applies only when |r| < 1. If |r| ≥ 1, the series diverges.
For geometric series, check |r| < 1. For general series, use convergence tests: ratio test, root test, integral test, comparison test, and limit comparison test. A series converges if its sequence of partial sums approaches a finite limit.
First term a = 72, common ratio r = 60/72 = 5/6. Since |5/6| < 1, the series converges. Sum = 72/(1−5/6) = 72/(1/6) = 432.
No, infinite arithmetic series always diverge because their terms do not approach zero. Only geometric series with |r| < 1, telescoping series, and certain other specialized series can converge to a finite sum.
Several infinite series approximate π. The Gregory-Leibniz series: π/4 = 1 − 1/3 + 1/5 − 1/7 + … converges slowly. More efficient series like the Ramanujan-Sato series converge rapidly and are used in modern π-calculation records.

Frequently Asked Questions About Infinite Series

This calculator specializes in infinite geometric series, which are the most common type encountered in algebra, calculus, and real-world applications. You can enter the first term and common ratio directly, or provide the first two terms to auto-detect the ratio.
A divergent series does not approach a finite limit. Its partial sums either grow without bound, oscillate indefinitely, or fail to settle on any specific value. For geometric series, divergence occurs when |r| ≥ 1. The calculator will clearly indicate when a series diverges.
The calculator accepts decimal inputs. For exact fractional results, you can enter decimal equivalents (e.g., 0.8333 for 5/6). The step-by-step breakdown displays the formula with your values so you can verify the calculation manually with fractions.
A sequence is an ordered list of numbers (e.g., 1, 1/2, 1/4, 1/8, …). An infinite series is the sum of all terms in an infinite sequence (e.g., 1 + 1/2 + 1/4 + 1/8 + … = 2). The series is the summation; the sequence is the list of terms.
The calculator uses JavaScript's native floating-point arithmetic with precision up to 6 decimal places for display. The geometric series formula is mathematically exact when |r| < 1. Results are displayed rounded for readability while maintaining computational accuracy.
A telescoping series is one where most terms cancel out in the partial sum, leaving only a few terms. This calculator focuses on geometric series. For telescoping series, you would need to identify the pattern of cancellation manually and compute the limit of the remaining terms.

Infinite Series Glossary

Geometric Series

A series where each term is multiplied by a constant ratio r. The infinite sum converges to a/(1−r) when |r| < 1.

Common Ratio (r)

The constant factor between consecutive terms in a geometric series: r = an+1/an.

Convergence

A series converges if its sequence of partial sums approaches a finite limit as more terms are added.

Divergence

A series diverges if its partial sums do not approach any finite limit; they may grow without bound or oscillate.

Partial Sum (Sn)

The sum of the first n terms of a series. For geometric series: Sn = a(1−rn)/(1−r).

Sigma Notation (Σ)

A compact way to write series: Σn=0∞ arn represents an infinite geometric series.

Ratio Test

A convergence test: if lim |an+1/an| < 1, the series converges absolutely; if > 1, it diverges.

Taylor Series

An infinite series that represents a function as a sum of terms calculated from its derivatives at a single point.

Editorial Review & Methodology

This infinite series calculator was built and reviewed by the NumbrWiz Editorial Team. The geometric series sum formula is a foundational concept in calculus and algebra, verified against standard mathematics curricula including AP Calculus BC, college-level calculus textbooks (Stewart, Larson), and online resources such as Wolfram MathWorld and Khan Academy.

  • Formula verification: Cross-checked against multiple authoritative calculus and algebra sources including Stewart's Calculus: Early Transcendentals.
  • Edge case testing: Tested with |r| close to 1, negative ratios, zero first term, and values near floating-point precision limits.
  • UX review: Designed for intuitive input with clear convergence/divergence indicators and step-by-step breakdown.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational purposes; verify critical calculations independently.

Page last reviewed: May 2026 · NumbrWiz Editorial Team