Interval of Convergence Calculator

📏 Interval of Convergence Calculator

Enter coefficients of the power series (comma separated) ✍️ Enter the center (c) of the series 📍 🔎 Calculate Interval of Convergence

📘 What is an Interval of Convergence?

The Interval of Convergence of a power series is the range of x-values for which the series converges to a finite number. Outside this interval, the series diverges. The interval depends on the Radius of Convergence and the behavior of the series at its endpoints.

⚙️ Formula Used by the Calculator

• Radius of Convergence: R = lim_n→∞ |aₙ / aₙ₊₁|
• Interval: (c – R, c + R)

🧪 How to Test the Endpoints

1. Substitute each endpoint value (x = c – R and x = c + R) into the original series.
2. Use the Ratio Test, Root Test, or Alternating Series Test to check convergence.
3. If the series converges at the endpoint, include it in the interval using brackets [ ].
4. If it diverges, exclude it using parentheses ( ).

🎓 Example

Consider the series Σ (xⁿ / n²). The radius of convergence is R = 1, and c = 0. The interval of convergence is (−1, 1). Test endpoints:

• x = 1 → series becomes Σ (1/n²), converges ✅
• x = −1 → series becomes Σ ((−1)ⁿ / n²), converges ✅

Therefore, the full interval is [−1, 1].

❓ Frequently Asked Questions (FAQs)

1. What is the interval of convergence in a power series?

It is the set of all x-values where the series converges to a finite number.

2. How do you find the interval of convergence?

Find the radius of convergence (R) and then compute (c – R, c + R). Test both endpoints to check convergence behavior.

3. What happens outside the interval?

The power series diverges for all x outside the interval of convergence.

4. Why are endpoint tests important?

Because the series may converge inside the interval but behave differently at the boundaries.

5. Is the interval always symmetric around the center?

Yes, the interval extends equally on both sides of the center, but endpoint behavior can differ.

🏁 Conclusion

The Interval of Convergence Calculator helps you quickly determine where your power series converges and understand endpoint behavior. It’s a vital learning and computation tool for students, teachers, and mathematicians working with power series and infinite sums. For deeper understanding, explore our related tool — the Radius of Convergence Calculator. 🚀