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The Maclaurin series of the logarithm function (+) is conditionally convergent for x = 1. The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
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.
Absolute convergence over sets. A generalization of the absolute convergence of a series, is the absolute convergence of a sum of a function over a set. We can first consider a countable set and a function We will give a definition below of the sum of over written as.
Definition. We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below). Suppose is a set and is a sequence of real-valued functions on it. We say the sequence is uniformly convergent on with limit if for every ...
Definition. More precisely, a series of real numbers is said to converge conditionally if exists (as a finite real number, i.e. not or ), but. A classic example is the alternating harmonic series given by which converges to , but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series ...
The Riemann zeta function is defined for real > by the convergent series = = = + + +, which for = would be the harmonic series. It can be extended by analytic continuation to a holomorphic function on all complex numbers except x = 1 {\displaystyle x=1} , where the extended function has a simple pole .
v. t. e. In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862. [1]
has radius of convergence 1 and converges everywhere on the boundary absolutely. If h is the function represented by this series on the unit disk, then the derivative of h(z) is equal to g(z)/z with g of Example 2. It turns out that h(z) is the dilogarithm function. Example 4: The power series