Taylor series

A Taylor series is a power series whose coefficients are determined by a function’s derivatives at a specified point. Where it converges to the function, it provides a representation of that function.

Taylor polynomial: A finite sum of terms from a Taylor series, used to approximate a function near its expansion point. Truncating the series gives the local polynomial approximation used in calculations.
Derivative: A measure of a function’s instantaneous rate of change, defined through a limit of difference quotients. Each Taylor coefficient uses a derivative evaluated at the expansion point.
Small-angle approximation: Approximations of trigonometric functions for angles near zero, such as sin x ≈ x. The sine Taylor series explains why these approximations work and how their errors grow.
Taylor's theorem without a remainder estimate: A finite Taylor expansion statement that does not by itself guarantee the remainder vanishes as the degree increases. Having Taylor polynomials of every degree does not ensure their series converges to the function.
Taylor's theorem: A theorem expressing a function as a finite Taylor polynomial plus a remainder under suitable differentiability conditions. Its remainder term quantifies the error when a Taylor series is truncated.
Factorial: For a nonnegative integer n, the product of all positive integers up to n, with 0! equal to 1. Dividing the nth derivative by n! gives the standard Taylor coefficient.
Numerical analysis: The study of algorithms for obtaining approximate numerical solutions to mathematical problems. Taylor polynomials support error estimates and local approximation methods in computation.
Smooth function: A function with continuous derivatives of every nonnegative integer order. Smoothness alone does not guarantee equality with the function’s Taylor series.
Radius of convergence: The distance from a power series’ center within which the series converges. It specifies how far from the expansion point the Taylor series can converge.
Power series: An infinite series whose terms are constants multiplied by nonnegative integer powers of a variable. Taylor series are power series with coefficients fixed by derivatives.

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