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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 mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
This is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form. The Lagrange form of the remainder is found by choosing G ( t ) = ( x − t ) k + 1 {\displaystyle G(t)=(x-t)^{k+1}} and the Cauchy form by choosing G ( t ) = t − a {\displaystyle G(t)=t-a} .
For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc x (while for not too large values of x, the above Taylor expansion at 0 provides a very fast convergence).
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
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 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 formula for the exponential results from reducing the powers of G in the series expansion and identifying the respective series coefficients of G 2 and G with −cos(θ) and sin(θ) respectively. The second expression here for e Gθ is the same as the expression for R ( θ ) in the article containing the derivation of the generator , R ( θ ...