Maclaurin Series

A Maclaurin series is a special case of the Taylor series where the infinite sum of derivatives is evaluated at the point c=0. In other words, a Maclaurin series is a power series expansion of a function around the origin. It is named after the Scottish mathematician Colin Maclaurin, who contributed significantly to the development of this concept.

The Maclaurin series of a function f(x) can be expressed as f(x) = ∑(f^n(0)/n!)(x^n), where the summation is taken over all non-negative integers n, and f^n(0) denotes the nth derivative of the function evaluated at x=0. Maclaurin series are particularly useful for approximating functions that are smooth and have continuous derivatives at the origin.

Examples of common Maclaurin series include those for the exponential function, sine, cosine, and natural logarithm. Maclaurin series are widely used in calculus, mathematical analysis, and applied mathematics to study the behavior of functions and their derivatives, as well as to solve differential equations and model various phenomena.