Maclaurin Series

Definition

A Maclaurin Series is a specific type of Taylor series where the center of expansion is at the origin, $x = 0$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$

• How to read: “The function f of x equals the sum from n equals zero to infinity of the n-th derivative of f evaluated at zero, divided by n factorial, all times x raised to the n-th power.”
• Meaning / when to use: Use to approximate a differentiable function locally near $x=0$. It is algebraically simpler than a general Taylor series because evaluating derivatives at zero is often trivial.

Why It Matters

Maclaurin series provide standardized polynomial representations for the most fundamental functions in mathematics and science (such as $e^x$, $\sin x$, $\cos x$, and $\frac{1}{1-x}$). These representations allow for rapid computation, simple limit evaluations (via L’Hôpital’s rule replacements), and quick approximations in physics near the origin (e.g. small-angle approximation $\sin \theta \approx \theta$).

Core Concepts

• Expansion at the Origin: Centering the approximation at $x = 0$ simplifies the terms, eliminating the $(x - a)$ subtraction.
• Common Series Expansions:
  • Exponential: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
  • Sine: $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
  • Cosine: $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$
• Convergence at Zero: The series is guaranteed to converge at least at the point $x = 0$.

Connected Concepts

• Taylor Series
• Taylor’s Formula for Two Variables
• Fundamental Theorem of calculus