Taylor Series Calculator
Understanding Taylor Series
What is a Taylor Series?
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point.
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...
= Σ(n=0 to ∞) [f⁽ⁿ⁾(a)(x-a)ⁿ/n!]
where:
- f⁽ⁿ⁾(a) is the nth derivative at x=a
- a is the center point
- n! is factorial of n
Types of Series Expansions
Maclaurin Series
- Special case where a=0
- Simplified calculations
- Common for elementary functions
Power Series
- General form of series
- Infinite polynomial
- Convergence radius
Laurent Series
- Includes negative powers
- Complex analysis
- Pole behavior
Convergence Properties
Radius of Convergence
- Maximum valid distance
- Determined by singularities
- Ratio test application
Error Bounds
- Lagrange remainder
- Truncation error
- Error estimation
Uniform Convergence
- Equal convergence rate
- Term-by-term operations
- Weierstrass M-test
Applications
Numerical Methods
- Function approximation
- Integration techniques
- Differential equations
Physics Applications
- Wave equations
- Quantum mechanics
- Perturbation theory
Engineering Uses
- Signal processing
- Control systems
- Error analysis