🧮 Power Series Representation Calculator
📘 What is a Power Series Representation?
A power series representation expresses a function as an infinite sum of powers of (x – a), where a is the expansion point. This helps approximate complex functions using polynomials for easier computation and analysis.
🧩 Connection to Radius of Convergence
The power series converges only within a certain interval around the expansion point. This interval is determined by the radius of convergence. You can easily calculate this using our Radius of Convergence Calculator.
⚙️ How to Use the Power Series Representation Calculator?
- Enter your desired function, e.g.,
ln(1+x)ore^x. - Choose the expansion point (a).
- Select the number of terms you want to include.
- Click Calculate Power Series to generate the polynomial representation.
🎓 Example
For f(x) = sin(x) expanded about a = 0, the series is: f(x) = x − x³/3! + x⁵/5! − x⁷/7! + …
✨ Benefits of Using This Tool
- Generates precise power series expansions instantly ⚡
- Ideal for calculus, mathematical modeling, and analysis 📘
- Educational visualization of polynomial approximations 📈
- Includes convergence context for better understanding ✅
❓ FAQs
1. What does a power series represent?
It’s a polynomial approximation of a function that converges to the function within a certain interval.
2. What’s the difference between Taylor and Power Series?
Every Taylor series is a power series, but not all power series are derived from a function’s derivatives.
3. What defines the convergence region?
The radius of convergence defines where the series accurately represents the function.
4. Can I change the expansion point?
Yes, you can set any point “a” for expansion — not just 0 like in the Maclaurin Series.
🏁 Conclusion
The Power Series Representation Calculator allows you to express functions as polynomial sums centered around any point. Combine it with the Radius of Convergence Calculator to fully understand where your series converges. 🚀