Convergence Calculator

Geometric Series Convergence Calculator

Calculation Results

function calculateConvergence() { var a = parseFloat(document.getElementById('firstTerm').value); var r = parseFloat(document.getElementById('commonRatio').value); var resultDiv = document.getElementById('convergenceResult'); var statusDiv = document.getElementById('statusResult'); var sumDiv = document.getElementById('sumResult'); var formulaDiv = document.getElementById('formulaDisplay'); if (isNaN(a) || isNaN(r)) { alert("Please enter valid numerical values for both the first term and the common ratio."); return; } resultDiv.style.display = "block"; var absoluteR = Math.abs(r); if (absoluteR < 1) { var sum = a / (1 – r); statusDiv.innerHTML = "Status: The series CONVERGES."; sumDiv.innerHTML = "Sum of Infinite Series (S∞): " + sum.toLocaleString(undefined, {maximumFractionDigits: 4}); formulaDiv.innerHTML = "Used Formula: S = a / (1 – r) because |" + r + "| < 1"; } else { statusDiv.innerHTML = "Status: The series DIVERGES."; sumDiv.innerHTML = "Sum: Does not exist (approaches ±Infinity)"; formulaDiv.innerHTML = "Reason: A geometric series diverges when the absolute value of the common ratio |r| ≥ 1."; } }

---

Understanding Convergence in Geometric Series

In mathematics, convergence refers to the property of an infinite series where the sum of its terms approaches a specific, finite limit as the number of terms increases. This calculator specifically focuses on Geometric Series, which are sequences 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).

The Geometric Series Formula

A geometric series is represented as: a + ar + ar² + ar³ + ...

Where:

• a is the first term.
• r is the common ratio.

How to Test for Convergence

The behavior of an infinite geometric series is determined entirely by the absolute value of the common ratio |r|:

1. Convergent: If |r| < 1, the series converges to a finite sum. The formula used is S = a / (1 - r).
2. Divergent: If |r| ≥ 1, the terms do not get smaller fast enough (or they get larger), and the sum does not settle on a single value. It is said to diverge to infinity or oscillate.

Real-World Example

Imagine you have a series where the first term a = 10 and the common ratio r = 0.5.

• Term 1: 10
• Term 2: 5
• Term 3: 2.5
• Term 4: 1.25

Since |0.5| < 1, the series converges. Using the formula: 10 / (1 - 0.5) = 10 / 0.5 = 20. The sum of all infinite terms will never exceed 20.

Applications of Convergence

Convergence testing is vital in various fields, including:

• Physics: Determining the stability of systems and calculating total displacement in bouncing objects.
• Finance: Calculating the present value of a perpetuity or valuing stocks with the Gordon Growth Model.
• Engineering: Signal processing and control systems analysis where infinite responses must be bounded.
• Computer Science: Analyzing the complexity of recursive algorithms and fractal geometry.