Taylorpolinom

Taylorpolinom (Taylor polynomial) is a polynomial that provides a local approximation to a function around a point a. It is constructed from the function’s derivatives at a, capturing the behavior of the function up to a chosen order. The degree-n Taylor polynomial of f about a is

T_n(x) = sum_{k=0}^n f^(k)(a) / k! * (x − a)^k.

If a = 0, the polynomial is called the Maclaurin polynomial.

Remainder and error: The difference between f and its Taylor polynomial, R_n(x) = f(x) − T_n(x), measures the approximation error. Under suitable smoothness, the remainder can be written in Lagrange form as

R_n(x) = f^(n+1)(ξ) / (n+1)! * (x − a)^(n+1)

for some ξ between a and x. This form provides an estimate of the error for x near a. If the derivatives satisfy stronger conditions, the Taylor series, the infinite sum of polynomials, may converge to f in a neighborhood of a.

Convergence and variants: If f is analytic at a, the Taylor series converges to f in some interval about a. There is also a multivariate version for functions of several variables. The Maclaurin series is the special case around a = 0. In practice, Taylor polynomials are used for local approximations, numerical methods, and the analysis of function behavior near a point.

History: The concept is named after Brook Taylor, who introduced it in the early 18th century.