Taylor's theorem

Taylor's theorem expresses a sufficiently differentiable function near a point as a finite polynomial built from its derivatives there, plus a remainder that measures the error.

Taylor polynomial: A polynomial whose coefficients are determined by a function’s derivatives at a specified point. It is the finite expression that Taylor’s theorem pairs with a remainder.
Derivative: A limit describing a function’s instantaneous rate of change or local linear behavior. Taylor coefficients use successive derivatives to encode local behavior.
Linearization: Approximation of a function near a point by its tangent line. It is the first-degree Taylor approximation and underlies local error estimates.
Laurent series: A series expansion in integer powers that may include negative powers of the variable. It handles isolated singularities where an ordinary Taylor expansion may fail.
Lagrange form of the Taylor remainder: A remainder formula expressing the error through a higher derivative at an intermediate point. It turns the approximation error into a bound when the higher derivative is controlled.
Higher derivative: A derivative obtained by differentiating a function repeatedly. The theorem uses derivatives of several orders to build its polynomial and control its remainder.
Newton's method: An iterative method for approximating a root by repeatedly using tangent-line intersections. Its local convergence analysis uses Taylor expansions around the root.
Fourier series: An expansion of a periodic function in sines, cosines, or complex exponentials. It represents global periodic behavior rather than local behavior from derivatives at one point.
Integral form of the Taylor remainder: A formula expressing Taylor’s remainder as an integral involving a higher derivative. It derives the error from accumulated variation beyond the polynomial’s degree.
Factorial: For a nonnegative integer, the product of all positive integers up to that integer, with zero factorial equal to one. Factorials normalize the derivative coefficients in the Taylor polynomial.

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