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In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715.
For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. [1] There are several versions of Taylor's theorem, some ...
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref. [3] Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method. [4]
This formula can be obtained by Taylor series expansion: (+) = + ′ ()! ″ ()! () +. The complex-step derivative formula is only valid for calculating first-order derivatives. A generalization of the above for calculating derivatives of any order employs multicomplex numbers , resulting in multicomplex derivatives.
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case = states that = + ′ () + where is the remainder term. The linear approximation is obtained by dropping the remainder: () + ′ ().
A function is analytic if and only if for every in its domain, its Taylor series about converges to the function in some neighborhood of . This is stronger than merely being infinitely differentiable at x 0 {\displaystyle x_{0}} , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is ...
The function () = (/) is the uniform limit of its Taylor expansion, which starts with degree 3. Also, ‖ f − g ‖ ∞ < c {\displaystyle \|f-g\|_{\infty }<c} . Thus to ϵ {\displaystyle \epsilon } -approximate f ( x ) = x {\displaystyle f(x)=x} using a polynomial with lowest degree 3, we do so for g ( x ) {\displaystyle g(x)} with c ...