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Radius of convergence. In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal ...
t. e. In mathematical analysis, particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations of how quickly that sequence approaches its limit. These are broadly divided into rates and orders of convergence that describe how quickly a sequence ...
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix. This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C). In this case, for the graph G define:
hide. 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 relatively unknown until Hadamard rediscovered it. [ 2 ]
The radius of this disc is known as the radius of convergence, and can in principle be determined from the asymptotics of the coefficients a n. The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.
Geometrically, the two Laurent series may have non-overlapping annuli of convergence. Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at , and inner radius of convergence 0, so they both converge on an overlapping annulus.
The radius of convergence is 1 / e , as may be seen by the ratio test. The function defined by this series can be extended to a holomorphic function defined on all complex numbers with a branch cut along the interval (−∞, − 1 / e ]; this holomorphic function defines the principal branch of the Lambert W function.
A hypergeometric series is formally defined as a power series. in which the ratio of successive coefficients is a rational function of n. That is, where A (n) and B (n) are polynomials in n. For example, in the case of the series for the exponential function, we have: So this satisfies the definition with A(n) = 1 and B(n) = n + 1.