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  2. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

    It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by Brook Taylor, [8] after whom the series are now named. The Maclaurin series was named after Colin Maclaurin, a Scottish mathematician, who published a special case of the Taylor result in the mid-18th century.

  3. Euler–Maclaurin formula - Wikipedia

    en.wikipedia.org/wiki/Euler–Maclaurin_formula

    For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.

  4. Power series - Wikipedia

    en.wikipedia.org/wiki/Power_series

    Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center c is equal to zero, for instance for Maclaurin series.

  5. Cumulant - Wikipedia

    en.wikipedia.org/wiki/Cumulant

    More generally, any coefficient of the Maclaurin series can also be expressed in terms of mixed moments, although there are no concise formulae. Indeed, as noted above, one can write it as a joint cumulant by repeating random variables appropriately, and then apply the above formula to express it in terms of mixed moments.

  6. Real analysis - Wikipedia

    en.wikipedia.org/wiki/Real_analysis

    In the case that a = 0, the series is also called a Maclaurin series. 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 ...

  7. Binomial series - Wikipedia

    en.wikipedia.org/wiki/Binomial_series

    where the power series on the right-hand side of is expressed in terms of the (generalized) binomial coefficients ():= () (+)!.Note that if α is a nonnegative integer n then the x n + 1 term and all later terms in the series are 0, since each contains a factor of (n − n).

  8. Small-angle approximation - Wikipedia

    en.wikipedia.org/wiki/Small-angle_approximation

    The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation , cos ⁡ θ {\displaystyle \textstyle \cos \theta } is approximated as either 1 {\displaystyle 1} or as 1 − 1 2 θ 2 {\textstyle 1-{\frac {1}{2}}\theta ^{2}} .

  9. 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.