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  2. 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.

  3. Geometric series - Wikipedia

    en.wikipedia.org/wiki/Geometric_series

    The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .

  4. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_function

    In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.

  5. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

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

  6. 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.

  7. Generalized hypergeometric function - Wikipedia

    en.wikipedia.org/wiki/Generalized_hypergeometric...

    Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D. In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n.

  8. Series (mathematics) - Wikipedia

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

    Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions. The Hilbert–Poincaré series is a formal power series used to study graded algebras.

  9. Generating function transformation - Wikipedia

    en.wikipedia.org/wiki/Generating_function...

    Series multisection provides formulas for generating functions enumerating the sequence {+} given an ordinary generating function () where ,, , and <.In the first two cases where (,):= (,), (,), we can expand these arithmetic progression generating functions directly in terms of ():