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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.
In particular, the Taylor expansion holds in the form = + (), = = ()! (), where the remainder term R k is complex analytic. Methods of complex analysis provide some powerful results regarding Taylor expansions.
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.
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The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. [1]
In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: [1]
Thus to -approximate () = using a polynomial with lowest degree 3, we do so for () with < / by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of g ( x ) {\displaystyle g(x)} , obtaining an approximation of lowest degree 9, 27, 81...
If the analytic function f has the Taylor expansion = + + + then a matrix function () can be defined by substituting x by a square matrix: powers become matrix powers, additions become matrix sums and multiplications by coefficients become scalar multiplications.