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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.
The exponential function (in blue), and the sum of the first n + 1 terms of its power series (in red) The exponential function is the sum of a power series: [2] [3] = + +! +! + = =!, where ! is the factorial of n (the product of the n first positive integers).
which are particularly useful in the cases where the component sequence generating functions, (), can be expanded in a Laurent series, or fractional series, in , such as in the special case where all of the component generating functions are rational, which leads to an algebraic form of the corresponding diagonal generating function.
A power series with coefficients in the field of algebraic numbers = =! ¯ [[]]is called an E-function [1] if it satisfies the following three conditions: . It is a solution of a non-zero linear differential equation with polynomial coefficients (this implies that all the coefficients c n belong to the same algebraic number field, K, which has finite degree over the rational numbers);
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.
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent. The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". [1]
There exist many types of convergence for a function series, such as uniform convergence, pointwise convergence, and convergence almost everywhere.Each type of convergence corresponds to a different metric for the space of functions that are added together in the series, and thus a different type of limit.
When the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series. In general, for any infinite sequence a i , the following power series identity holds: