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The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations. The Taylor approximations for ln(1 + x) (black). For x > 1, the approximations diverge. Pictured is an accurate approximation of sin x around the point x = 0. The ...
This function was plotted above to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large. This kind of behavior is easily understood in the framework of complex analysis.
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.
The Taylor expansion would be: + where / denotes the partial derivative of f k with respect to the i-th variable, evaluated at the mean value of all components of vector x. Or in matrix notation , f ≈ f 0 + J x {\displaystyle \mathrm {f} \approx \mathrm {f} ^{0}+\mathrm {J} \mathrm {x} \,} where J is the Jacobian matrix .
The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.
In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: [1] = + + = = + +. This series converges in the complex disk | |, except for = (where =).
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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]