Maclaurin Series
A Maclaurin series expands a function as an infinite polynomial centered at zero.
Learn the formula with examples for common functions.
The Formula
A Maclaurin series is a special case of the Taylor series, centered at x = 0. It represents a function as an infinite sum of terms calculated from the derivatives of the function at zero.
This series is named after Scottish mathematician Colin Maclaurin, who popularized the technique in the 18th century, although the concept was known earlier. The Maclaurin series is extremely useful for approximating functions when x is near zero, and many common functions have well-known expansions.
Key Maclaurin series include: eˣ = 1 + x + x²/2! + x³/3! + ... (converges for all x). sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ... (converges for all x). cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ... (converges for all x). 1/(1-x) = 1 + x + x² + x³ + ... (converges for |x| < 1).
Variables
| Symbol | Meaning |
|---|---|
| f(x) | The function being expanded |
| f(0), f'(0), f''(0), ... | The function and its derivatives evaluated at x = 0 |
| n! | Factorial of n |
| x | The variable (series converges best when x is near 0) |
Example 1
Use the first 4 terms of the Maclaurin series for eˣ to approximate e^0.5.
The series: eˣ ≈ 1 + x + x²/2! + x³/3!
Substitute x = 0.5: 1 + 0.5 + 0.25/2 + 0.125/6
= 1 + 0.5 + 0.125 + 0.02083
e^0.5 ≈ 1.6458 (actual value: 1.6487)
Example 2
Use the first 3 non-zero terms of the Maclaurin series for sin(x) to approximate sin(0.3).
The series: sin(x) ≈ x - x³/3! + x⁵/5!
Substitute x = 0.3: 0.3 - 0.027/6 + 0.00243/120
= 0.3 - 0.0045 + 0.00002025
sin(0.3) ≈ 0.29552 (actual value: 0.29552)
When to Use It
Maclaurin series are foundational in calculus and applied mathematics.
- Approximating function values without a calculator
- Evaluating limits that are otherwise indeterminate
- Solving differential equations with power series methods
- Computer algorithms for calculating trigonometric and exponential functions