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Power series. In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the n th term and c is a constant called the center of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions.
Furthermore, even if the series converges everywhere on the boundary (even uniformly), it does not necessarily converge absolutely. Example 1: The power series for the function f(z) = 1/(1 − z), expanded around z = 0, which is simply =, has radius of convergence 1 and diverges at every point on the boundary. Example 2: The power series for g ...
The negative binomial series includes the case of the geometric series, the power series [1] = = (which is the negative binomial series when =, convergent in the disc | | <) and, more generally, series obtained by differentiation of the geometric power series: = ()! with =, a positive integer.
A theorem that determines the radius of convergence of a power series. In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, [1] but remained ...
An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence. This is because a power series is absolutely convergent on the interior of its disk of convergence. [a]
Many other power series can be written as transformations and combinations of geometric series, making the geometric series formula a convenient tool for calculating formulas for those power series as well. [13] [14] As a power series, the geometric series has a radius of convergence of 1. [15]
t. e. In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
t. e. In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.