🔢 Taylor Polynomial Approximation Calculator
📘 What is the Taylor Polynomial Approximation Calculator?
The Taylor Polynomial Approximation Calculator builds the degree-(n−1) Taylor polynomial for a function \(f(x)\) about a point \(a\): \[ P_{n-1}(x)=\sum_{k=0}^{n-1}\frac{f^{(k)}(a)}{k!}(x-a)^k \] This is the practical polynomial used in numerical approximations for physics, engineering, and analysis.
🔗 Relation to Radius of Convergence
Every Taylor series has a radius of convergence that limits where the infinite series equals the function. Use our Radius of Convergence Calculator to investigate that radius for your function.
🧪 Example
For \(f(x)=\mathrm{e}^x\) about \(a=0\) with \(n=5\): \(P_4(x)=1 + x + x^2/2! + x^3/3! + x^4/4!\). Use the tool to compute the polynomial and evaluate numerical error for any x.
❓ FAQs
Q: Can I use trig, exponential, and log functions?
A: Yes — common functions like sin(x), cos(x), e^x, ln(1+x) are supported. Use mathjs notation if needed (e.g., log(1+x) for natural log).
Q: What if the Taylor series doesn’t converge at my x?
A: The Taylor polynomial still gives a local approximation; check the radius of convergence to know where the infinite series is valid.