Convergence Test Calculator

📊 Convergence Test Calculator

Determine whether an infinite series converges or diverges using Ratio, Root, and Comparison Tests.

Enter the nth-term of the series (a_n): 🔍 Test Convergence

📘 What is a Convergence Test?

A Convergence Test determines whether an infinite series ∑ aₙ converges to a finite value or diverges. These tests are crucial for understanding where a power series is valid — a concept directly linked to the Radius of Convergence Calculator.

⚙️ Types of Convergence Tests

1. Ratio Test

Uses the limit of |a_n+1 / a_n| as n → ∞. – If < 1 → Series converges. – If > 1 → Diverges. – If = 1 → Inconclusive.

2. Root Test

Uses the limit of |a_n|^1/n as n → ∞. – If < 1 → Converges. – If > 1 → Diverges. – If = 1 → Inconclusive.

3. Comparison Test

Compares the series to a known benchmark series (like p-series). If the test series is smaller than a convergent benchmark → it also converges.

🧩 How This Tool Works

1. Enter the general term a_n of your series (e.g., (1/n²), (3ⁿ/n!)).
2. Click Test Convergence.
3. The calculator applies the Ratio and Root Tests to determine convergence.

📈 Example

For aₙ = 1/n²: – Ratio Test → 0.999 (Convergent) ✅ – Root Test → 1 (Inconclusive but Comparison Test shows Convergent)

🏁 Conclusion

The Convergence Test Calculator helps determine whether a series converges or diverges. To find the exact range where a power series converges, visit the Radius of Convergence Calculator.