Maclaurin Series Fundamentals: Efficient Approximations for Common Functions

The Maclaurin series is a specific Taylor series that expands a function around \( x = 0 \). It expresses functions as infinite polynomials using derivatives evaluated at zero, which is especially useful for approximating functions near zero. The general form of a Maclaurin series for a function \( f(x) \) is:

\( [
f(x) = f(0) + f'(0)x + \frac{f”(0)}{2!}x^2 + \frac{f”'(0)}{3!}x^3 + \ldots
] \)

Each term’s coefficient involves a higher derivative of \( f(x) \) at \( x = 0 \), divided by the factorial of the term’s order. Common examples include \( e^x \), \( \sin(x) \), and \( \cos(x) \), which have useful Maclaurin series expansions:

• For \( e^x \): \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \)
• For \( \sin(x) ): ( x – \frac{x^3}{3!} + \frac{x^5}{5!} – \ldots \)
• For \( \cos(x) ): ( 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \ldots \)

These expansions are widely used in physics, engineering, and computational science for simplifying complex function evaluations near zero.