📈 Series Absolute Convergence Calculator

Determine whether an infinite series converges absolutely, conditionally, or diverges — with a clear, step-by-step explanation.

How to use this tool

  1. Enter your nth-term expression aₙ using variable n.
  2. Set number of terms to analyze for decreasing pattern.
  3. Choose a large n value for checking limₙ→∞ aₙ.
  4. Click Test Convergence to see detailed steps.

Understanding Absolute Convergence of a Series

A series ∑ aₙ is said to be absolutely convergent if ∑ |aₙ| converges. Absolute convergence guarantees the original series also converges.

🧮 Steps to Check Absolute Convergence

  1. Take the absolute value of each term, |aₙ|.
  2. Test whether the resulting positive-term series converges.
  3. If ∑ |aₙ| converges → the series converges absolutely.
  4. If ∑ |aₙ| diverges but ∑ aₙ converges → the series is conditionally convergent.

📘 Example

Consider ∑ (-1)ⁿ / n². Here, |aₙ| = 1/n² forms a p-series with p = 2, which converges. Hence, the series is absolutely convergent.

💡 Related Tool

Explore our main Radius of Convergence Calculator to understand how power and Taylor series behave within their radius of convergence.

✨ Benefits of Using This Tool

  • Automatic absolute value and limit evaluation.
  • Decreasing-term analysis with instant output.
  • Educational step-by-step breakdown.
  • Perfect for students learning about infinite series.

This Series Absolute Convergence Calculator is a simple yet powerful learning aid that bridges theory and computation. 🚀