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That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
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 iterative calculation of the above series, the following alternative formulation may be useful: = = (= (+)) = = + = because −(2k − 1)z 2 / k(2k + 1) expresses the multiplier to turn the k th term into the (k + 1) th term (considering z as the first term).
The Taylor series of f will converge in some interval in which all its derivatives are bounded and do not grow too fast as k goes to infinity. (However, even if the Taylor series converges, it might not converge to f, as explained below; f is then said to be non-analytic.) One might think of the Taylor series
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
Even if the PDF can be found, finding the moments (above) can be difficult. 4. The solution is to expand the function z in a second-order Taylor series; the expansion is done around the mean values of the several variables x. (Usually the expansion is done to first order; the second-order terms are needed to find the bias in the mean.
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.
That is, if we have a value for the cumulative distribution function, (), but do not know the x needed to obtain the (), we can use Newton's method to find x, and use the Taylor series expansion above to minimize the number of computations.