Search results
Results from the WOW.Com Content Network
The Taylor series of f converges uniformly to the zero function T f (x) = 0, which is analytic with all coefficients equal to zero. The function f is unequal to this Taylor series, and hence non-analytic. For any order k ∈ N and radius r > 0 there exists M k,r > 0 satisfying the remainder bound above.
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.
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: f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) . {\displaystyle f(x)\approx f(a)+f'(a)(x-a).}
A Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that | | < (the largest such R for which convergence is guaranteed is called the radius of convergence), or converge on the entire real line. Even a converging Taylor series may converge to a value different from the value of the function at ...
For every sequence α 0, α 1, α 2, . . . of real or complex numbers, the following construction shows the existence of a smooth function F on the real line which has these numbers as derivatives at the origin. [1] In particular, every sequence of numbers can appear as the coefficients of the Taylor series of a smooth function.
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]
After the example of a Taylor series not converging to the original function is introduced, the sentence "Now the estimates for the remainder for the Taylor polynomials show that the Taylor series of f converges uniformly to the zero function on the whole real axis" is in my opinion wrong, because it's not the estimates of the remainder term ...