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  2. Taylor's theorem - Wikipedia

    en.wikipedia.org/wiki/Taylor's_theorem

    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.

  3. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

    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 ...

  4. Linear approximation - Wikipedia

    en.wikipedia.org/wiki/Linear_approximation

    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).}

  5. Taylor expansions for the moments of functions of random ...

    en.wikipedia.org/wiki/Taylor_expansions_for_the...

    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.

  6. Series (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Series_(mathematics)

    A series of real or complex numbers is said to be ... is the Taylor series of ... who also gave a general form for the remainder of the Maclaurin formula. The most ...

  7. Non-analytic smooth function - Wikipedia

    en.wikipedia.org/wiki/Non-analytic_smooth_function

    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.

  8. Multi-index notation - Wikipedia

    en.wikipedia.org/wiki/Multi-index_notation

    Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

  9. First-order second-moment method - Wikipedia

    en.wikipedia.org/wiki/First-order_second-moment...

    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]