Taylor & Maclaurin Series Generator
Approximate functions with polynomials: Pn(x) ≈ f(x)
Configuration
a=0 for Maclaurin
Max 10 recommended
Approximation Polynomial Pn(x)
Convergence Visualization
Visual: See how the Red curve (approximation) matches the Blue curve (original) near the black dot.
Understanding Taylor Series
Many functions in math (like sin(x), ex, or ln(x)) are complicated to calculate directly. A Taylor Series is a way to approximate these complex functions using an infinite sum of simple polynomial terms $ (x, x^2, x^3, …) $.
The Formula
The series is built using derivatives at a single point a (the center).
f(x) ≈ f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f”(a)}{2!}(x-a)^2 + …
Taylor vs. Maclaurin
A Maclaurin Series is just a special nickname for a Taylor Series where the center point is a = 0. It is the most common form used in textbooks.
How it Works
Think of it as layering information:
- Term 0 (f(a)): Matches the Height of the function.
- Term 1 (f'(a)): Matches the Slope (direction).
- Term 2 (f”(a)): Matches the Curvature (concavity).
- Term 3 (f”'(a)): Matches the rate the curvature changes.
The more terms you add (higher “Order”), the further away from the center point the approximation remains accurate.
Frequently Asked Questions (FAQ)
Q: Why do calculators use this?
A: Computers can only really add, subtract, multiply, and divide. They can’t “do” geometry. To calculate sin(35°), your calculator actually uses a Taylor Series polynomial to estimate the value to high precision using just basic arithmetic.
Q: What is the “Radius of Convergence”?
A: For some functions, the Taylor Series only works near the center point a. If you go too far out, the approximation blows up and becomes useless. The distance from a where it is safe to use is the radius of convergence.
Q: Does order matter?
A: Yes! A higher order (n) means a better approximation that hugs the curve for a longer distance. A line (order 1) only touches; a parabola (order 2) curves with it; a cubic (order 3) wiggles with it.