Maclaurin Series
A Maclaurin series is a special case of the Taylor series where the expansion is centered at a = 0, providing a power series representation of a function in terms of its derivatives at the origin. It is particularly useful when a function and its derivatives are easy to evaluate at zero, yielding compact and elegant representations. The Maclaurin series underpins Euler's formula, connecting exponential and trigonometric functions through complex analysis.
Key Formula
f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …
LaTeX: f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots
| Symbol | Meaning | Unit |
|---|---|---|
| f^(n)(0) | nth derivative of f evaluated at x = 0 | varies |
| n! | Factorial of n (n! = 1×2×3×…×n) | dimensionless |
| x | Variable (series valid near x = 0) | dimensionless |
Worked Example
Problem
Derive the Maclaurin series for sin(x) up to the x⁵ term.
Solution
Step 1: Compute derivatives: f(x)=sin x, f'(x)=cos x, f''(x)=−sin x, f'''(x)=−cos x, f⁽⁴⁾(x)=sin x, f⁽⁵⁾(x)=cos x. Step 2: Evaluate at 0: f(0)=0, f'(0)=1, f''(0)=0, f'''(0)=−1, f⁽⁴⁾(0)=0, f⁽⁵⁾(0)=1. Step 3: Substitute: sin x = 0 + x + 0 − x³/6 + 0 + x⁵/120. Step 4: Result: sin x ≈ x − x³/6 + x⁵/120.
Answer
sin(x) ≈ x − x³/6 + x⁵/120 (first three non-zero terms)
Maclaurin Series for Key Functions
| Function | Maclaurin Series | First Nonzero Terms | Convergence |
|---|---|---|---|
| eˣ | Σ xⁿ/n! | 1 + x + x²/2 + x³/6 | All real x |
| sin x | Σ (−1)ⁿ x^(2n+1)/(2n+1)! | x − x³/6 + x⁵/120 | All real x |
| cos x | Σ (−1)ⁿ x^(2n)/(2n)! | 1 − x²/2 + x⁴/24 | All real x |
| ln(1+x) | Σ (−1)^(n+1) xⁿ/n | x − x²/2 + x³/3 | −1 < x ≤ 1 |
| arctan x | Σ (−1)ⁿ x^(2n+1)/(2n+1) | x − x³/3 + x⁵/5 | |x| ≤ 1 |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Named after Scottish mathematician Colin Maclaurin (1698–1746), who used this form extensively in his 1742 "Treatise of Fluxions". The series is a special case of the Taylor series; the name "Maclaurin series" acknowledges Maclaurin's systematic use of the a=0 expansion.