🧮 Taylor Series Expansion Calculator

📘 What is a Taylor Series Expansion Calculator?

The Taylor Series Expansion Calculator computes the polynomial approximation of a given function around a specified point (a). It expresses the function as an infinite sum of its derivatives at that point — providing a powerful way to approximate complex functions.

🧩 How Taylor Series Relates to the Radius of Convergence

Every Taylor series has a radius of convergence, which defines the interval within which the series converges to the function. Beyond this radius, the approximation no longer holds true. You can explore this concept in depth using our Radius of Convergence Calculator.

⚙️ How to Use the Taylor Series Expansion Calculator?

  1. Enter your desired function, e.g., sin(x) or e^x.
  2. Set the expansion point (a), commonly 0 for Maclaurin series.
  3. Choose how many terms you want to include in the expansion.
  4. Click Calculate Taylor Series to generate your polynomial approximation.

🎓 Example

For f(x) = ex expanded about a = 0 (Maclaurin Series): f(x) ≈ 1 + x + x²/2! + x³/3! + …

✨ Benefits of Using This Tool

  • Instantly expands functions into polynomial form ⚡
  • Useful for calculus, physics, and numerical methods 🧮
  • Educational for understanding function approximation 📘
  • Connects directly to convergence concepts 📈

❓ FAQs

1. What is a Taylor Series?

It’s a series that approximates a function as an infinite sum of derivatives evaluated at a point.

2. What is the difference between Taylor and Maclaurin Series?

A Maclaurin Series is just a Taylor Series centered at a = 0.

3. How does the radius of convergence affect it?

It defines the region where your Taylor Series accurately represents the function.

4. Can I expand trigonometric and logarithmic functions?

Yes, common functions like sin(x), cos(x), ex, and ln(1+x) are supported.

🏁 Conclusion

The Taylor Series Expansion Calculator makes it easy to visualize and compute polynomial approximations of any mathematical function. It’s an essential tool for calculus, numerical analysis, and understanding function behavior near a point. 🚀