Taylor Series

CORE DEFINITION

In mathematics, a Taylor series (also called Taylor expansion) represents a function as an infinite sum of terms, which are calculated from the values of the function's derivatives at a single point. It is named after the English mathematician Sir Brook Taylor, who published the Taylor formula and Taylor expansion in 1715. A Taylor series obtained by evaluating the derivatives at zero is called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. Lagrange was the first to propose the current form of Taylor's theorem with remainder, before 1797. In practical applications, Taylor series are truncated to a finite number of terms, and the error of such an approximation can be estimated using Taylor's theorem. A finite number of terms of a Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of its Taylor polynomials (if the limit exists). Even if the Taylor series converges at every point, the function and its Taylor series may not be equal.