Maclaurins

Maclaurins refers to the mathematical contributions of Colin Maclaurin, most notably the Maclaurin series and related results in calculus and analysis. The Maclaurin series is the Taylor series of a function expanded about zero and is written as f(x) = sum_{n=0}^\infty f^{(n)}(0)/n! x^n for functions that are analytic at 0. This form provides polynomial approximations to a function near x = 0, and the Maclaurin polynomial of degree n is the truncated series up to the n-th term.

Convergence and remainder: The series converges to f(x) for x within the function’s radius of convergence. If the function is sufficiently smooth, the error is captured by the remainder term, commonly R_n(x) = f^{(n+1)}(ξ)/((n+1)!) x^{n+1} for some ξ between 0 and x.

Examples: e^x has Maclaurin series 1 + x + x^2/2! + x^3/3! + ..., sin x = x - x^3/3! + x^5/5! - ..., cos x = 1 - x^2/2! + x^4/4! - ....

History and use: Colin Maclaurin developed these ideas in the 18th century. Maclaurin series are fundamental in numerical analysis and applied mathematics, enabling practical approximations, analysis of functions, and the solution of differential equations. The concept is a specific instance of the more general Taylor series expanded about zero.