Taylor Series
A Taylor series is an infinite power series representation of a smooth (infinitely differentiable) function, expanded about a point a, with coefficients determined by the function's derivatives at that point. It provides a polynomial approximation that converges to the original function within its radius of convergence. Taylor series are indispensable in numerical analysis, physics, and engineering for approximating complex functions, analyzing differential equations, and computing limits.
Key Formula
f(x) = Σ [f^(n)(a)/n!] · (x−a)^n, n from 0 to ∞
LaTeX: f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots
| Symbol | Meaning | Unit |
|---|---|---|
| f^(n)(a) | nth derivative of f evaluated at point a | varies |
| a | Center point of expansion | dimensionless |
| n! | Factorial of n | dimensionless |
| x | Variable | dimensionless |
Worked Example
Problem
Find the first four terms of the Taylor series for f(x) = eˣ centered at a = 0.
Solution
Step 1: Compute derivatives: f(x)=eˣ, f'(x)=eˣ, f''(x)=eˣ, f'''(x)=eˣ. Step 2: Evaluate at a=0: f(0)=1, f'(0)=1, f''(0)=1, f'''(0)=1. Step 3: Apply formula: eˣ = 1/0! + x/1! + x²/2! + x³/3! + … Step 4: Simplify: eˣ ≈ 1 + x + x²/2 + x³/6.
Answer
eˣ ≈ 1 + x + x²/2 + x³/6 (first four terms)
Common Taylor Series Expansions at a = 0
| Function | Series Expansion | Radius of Convergence | Common Application |
|---|---|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + … | ∞ (converges everywhere) | Exponential growth models |
| sin x | x − x³/3! + x⁵/5! − … | ∞ | Oscillation, wave equations |
| cos x | 1 − x²/2! + x⁴/4! − … | ∞ | Signal processing, optics |
| ln(1+x) | x − x²/2 + x³/3 − … | |x| < 1 | Logarithmic approximation |
| (1+x)^k | 1 + kx + k(k−1)x²/2! + … | |x| < 1 | Binomial approximation |
Interactive Tools
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Related Terms
Named after the English mathematician Brook Taylor (1685–1731), who published this result in 1715 in "Methodus Incrementorum Directa et Inversa". Earlier work by James Gregory and Johann Bernoulli anticipated the idea.