Compute the Maclaurin series (Taylor expansion about 0) of f(x) up to degree n.
A Maclaurin series is a Taylor series centered at 0: M(x) = Σ f⁽ᵏ⁾(0)/k! · xᵏ. Common Maclaurin series include e^x = Σ xᵏ/k!, sin(x) = x - x³/3! + x⁵/5! - ..., cos(x) = 1 - x²/2! + x⁴/4! - .... Coefficients are computed numerically via finite differences.
Maclaurin series
M(x) = Σ [f^(k)(0) / k!] · x^k, for k = 0 to n
Use Maclaurin when the function is well-behaved near 0. Use Taylor with a different center when you need accuracy far from the origin.
Functions with singularities or rapid oscillations near x=0 need many terms. The radius of convergence limits where the series is useful.