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  2. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

    The Taylor series of any polynomial is the polynomial itself.. The Maclaurin series of ⁠ 1 / 1x ⁠ is the geometric series + + + +. So, by substituting x for 1x, the Taylor series of ⁠ 1 / x ⁠ at a = 1 is

  3. Taylor's theorem - Wikipedia

    en.wikipedia.org/wiki/Taylor's_theorem

    Now its Taylor series centered at z 0 converges on any disc B(z 0, r) with r < |z − z 0 |, where the same Taylor series converges at z ∈ C. Therefore, Taylor series of f centered at 0 converges on B(0, 1) and it does not converge for any z ∈ C with |z| > 1 due to the poles at i and −i.

  4. Arctangent series - Wikipedia

    en.wikipedia.org/wiki/Arctangent_series

    In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: [1]

  5. Series expansion - Wikipedia

    en.wikipedia.org/wiki/Series_expansion

    A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.

  6. Universal Taylor series - Wikipedia

    en.wikipedia.org/wiki/Universal_Taylor_series

    Thus to -approximate () = using a polynomial with lowest degree 3, we do so for () with < / by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of g ( x ) {\displaystyle g(x)} , obtaining an approximation of lowest degree 9, 27, 81...

  7. Finite difference method - Wikipedia

    en.wikipedia.org/wiki/Finite_difference_method

    For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as (+) = + ′ ()! + ()! + + ()! + (),. Where n! denotes the factorial of n, and R n (x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function.

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  9. 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. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.