🔄 Alternating Series Convergence Test (Step-by-Step)

Enter the positive term aₙ for an alternating series of the form ∑ (-1)ⁿ aₙ. This calculator checks the Leibniz criteria and explains each step clearly.

How to use this tool

  1. Enter the nth-term expression aₙ using variable n.
  2. Set the number of terms to test for decreasing behavior (default is 12).
  3. Choose a large n value (default is 1000) for the limit test.
  4. Click Check Convergence to view step-by-step results.

Understanding the Alternating Series Convergence Test

The Alternating Series Test (also known as the Leibniz Criterion) determines whether an alternating series converges. A typical alternating series has terms that alternate in sign, such as:

∑ (-1)ⁿ aₙ or ∑ (-1)ⁿ⁺¹ aₙ, where aₙ > 0.

🧮 Conditions for Convergence

  1. The terms aₙ are positive.
  2. aₙ decreases monotonically (aₙ₊₁ ≤ aₙ).
  3. limₙ→∞ aₙ = 0.

If all these conditions hold, the alternating series converges.

📘 Example

Consider the series ∑ (-1)ⁿ / n. Here aₙ = 1/n decreases and tends to zero as n → ∞, so the series converges by the Leibniz test.

🔗 Related Tool

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

This tool is designed for students, engineers, and math enthusiasts who want to visualize convergence tests and learn step-by-step how alternating series behave.