Taylor polynomial

A finite polynomial built from a function’s derivatives at an expansion point. It approximates the function locally, with accuracy governed by the omitted terms and the function’s smoothness.

Taylor series: An infinite power series whose coefficients are determined by a function’s derivatives at an expansion point. The finite polynomial is obtained by truncating this series.
Derivative: The instantaneous rate of change of a function with respect to its input. Repeated derivatives supply the local data used in each coefficient.
Numerical analysis: The study of algorithms for approximating mathematical quantities with finite computation. Taylor polynomials provide local approximations used in numerical algorithms.
Padé approximant: A rational function chosen to match a specified number of terms in a power series. It can approximate functions better than a Taylor polynomial away from the expansion point.
Taylor's theorem: A theorem expressing a sufficiently differentiable function as a finite Taylor polynomial plus a remainder. It gives conditions and bounds for the error after truncation.
Factorial: The product of the positive integers up to a given nonnegative integer. Dividing the nth derivative by n! gives the standard Taylor coefficient.
Newton's method: An iterative algorithm that estimates a root by repeatedly using tangent-line approximations. Its update follows from a first-degree Taylor approximation.
Chebyshev approximation: Polynomial approximation designed to minimize or control error across an interval. Unlike Taylor approximation, it targets accuracy over a whole interval rather than near one point.
Lagrange form of the remainder: A representation of Taylor approximation error using a higher derivative at an intermediate point. It converts a bound on the next derivative into an explicit error estimate.
Power series: An infinite sum of powers of a variable multiplied by fixed coefficients. Taylor polynomials are finite versions of this broader series form.

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