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  2. 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 ...

  3. Taylor's theorem - Wikipedia

    en.wikipedia.org/wiki/Taylor's_theorem

    In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.

  4. Radius of convergence - Wikipedia

    en.wikipedia.org/wiki/Radius_of_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.

  5. Machin-like formula - Wikipedia

    en.wikipedia.org/wiki/Machin-like_formula

    Let be the amount of time spent on each digit (for each term in the Taylor series). The Taylor series will converge when: (()) = Thus: = ⁡ ⁡ For the first term in the Taylor series, all digits must be processed. In the last term of the Taylor series, however, there's only one digit remaining to be processed.

  6. Leibniz formula for π - Wikipedia

    en.wikipedia.org/wiki/Leibniz_formula_for_π

    The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...

  7. Arctangent series - Wikipedia

    en.wikipedia.org/wiki/Arctangent_series

    Each term of this modified series is a rational function with its poles at = in the complex plane, the same place where the arctangent function has its poles. By contrast, a polynomial such as the Taylor series for arctangent forces all of its poles to infinity.

  8. Residue (complex analysis) - Wikipedia

    en.wikipedia.org/wiki/Residue_(complex_analysis)

    Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz −1 is zero, and the integral is ...

  9. List of mathematical series - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_series

    An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.