Power Series Radius of Convergence Calculator

🔵 Power Series Radius of Convergence Calculator

Enter coefficients of the series (comma separated) ✍️ 📏 Calculate Radius of Convergence

📘 What is a Power Series Radius of Convergence Calculator?

The Power Series Radius of Convergence Calculator determines the distance within which a power series converges. In simple terms, it tells you how far you can move from the center of the series while it still produces valid results.

🧠 Understanding the Radius of Convergence

For a power series in the form Σ aₙ(x - c)ⁿ, the radius of convergence (R) is the value that defines where the series converges. The series converges when |x - c| < R and diverges when |x - c| > R.

⚙️ Formula Used

• Ratio Test: R = lim_n→∞ |aₙ / aₙ₊₁|
• Root Test: R = 1 / lim sup_n→∞ √[n]{|aₙ|}

💡 How to Use This Calculator

1. Enter the coefficients of your series separated by commas (e.g. 1, 1/2, 1/6).
2. Click the 📏 Calculate Radius of Convergence button.
3. The calculator will display the radius and the interval of convergence.

🎓 Example

Suppose your series is Σ (xⁿ / n!). Here, aₙ = 1/n! → aₙ / aₙ₊₁ = (1/n!) / (1/(n+1)!) = n+1. As n → ∞, the limit → ∞ → R = ∞. So the series converges for all x.

❓ Frequently Asked Questions (FAQs)

1. What does the radius of convergence tell us?

It shows the range of x values where a power series converges to a finite value.

2. Can the radius of convergence be infinite?

Yes. If a series converges for all real x, its radius of convergence is infinite (R = ∞).

3. What if my series has alternating or zero coefficients?

The calculator uses absolute values for ratio tests, so alternating signs are handled correctly.

4. What’s the difference between radius and interval of convergence?

The radius defines the distance around the center; the interval gives the exact range (c – R, c + R).

5. Is this calculator suitable for all power series?

Yes, it works for any numerical coefficient-based power series, as long as you provide enough terms.

🏁 Conclusion

The Power Series Radius of Convergence Calculator is a handy tool for students, teachers, and mathematicians who work with infinite series. It quickly determines where your series converges, saving you time and helping you understand function behavior more clearly. 🚀