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...

Absolute convergence at every point of the boundary: ∑ n = 1 ∞ z n n 2 {\textstyle \sum _{n=1}^{\infty }{\frac {z^{n}}{n^{2}}}} has radius of convergence 1 {\displaystyle...

_{k=0}^{\infty }(-1)^{k}(z-1)^{k}.} By the Cauchy–Hadamard theorem, its radius of convergence is 1. That is, f {\displaystyle f} is defined and analytic on the...

mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained...

plane. Bend radius Filling radius in Riemannian geometry Mean radius Radius of convergence Radius of convexity Radius of curvature Radius of gyration Semidiameter...

these have poles at c {\displaystyle c} , and inner radius of convergence 0, so they both converge on an overlapping annulus. Thus when defining formal...

not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The...

Domain of convergence of power series Riemann series theorem – Unconditionally convergent series converge absolutely Unconditional convergence – Order-independent...

a_{k}} with radius of convergence 1. {\displaystyle 1.} Suppose that the series ∑ k = 0 ∞ a k {\displaystyle \sum _{k=0}^{\infty }a_{k}} converges. Then G...

from it Radius of convergence (in calculus), the radius of the region where a complex power series converges Radius of curvature, a measure of how gently...

}c_{n}(z-a)^{n}} (this implies that the radius of convergence is positive). One of the most important theorems of complex analysis is that holomorphic functions...

an interpretation in terms of p-adic numbers: with an appropriate extension of the idea, the p-adic radius of convergence of the series is at least 1,...

is the principal part of f {\displaystyle f} at a {\displaystyle a} . If the Laurent series has an inner radius of convergence of 0 {\displaystyle 0} ...

that the power series has radius of convergence exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform...

whenever α {\displaystyle \alpha } is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (5), by comparison...

is expanded at one of its points of convergence to produce a power series with a larger radius of convergence. In effect, a set of data from a small region...

analytic within some radius of convergence; typically with a radius of convergence of | x − y | {\displaystyle |x-y|} . Thus, the ring of functions can be...

notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The...

x = ±i. This explains why the Taylor series of f(x) diverges for |x| > 1, i.e., the radius of convergence is 1 because the complexified function has a...

half-plane of convergence of a Dirichlet series are analogous to radius, boundary and disk of convergence of a power series. On the line of convergence, the...

series has a non-zero radius of convergence, i.e., g ( z ) {\displaystyle g(z)} represents an analytic function of z in a neighbourhood of z = f ( a ) . {\displaystyle...

Return of capital Return on capital Radius of curvature (optics) Receiver operating characteristic, ROC curve (statistics) Radius of convergence Rail operating...

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity lim sup n → ∞...

with its radius known as the radius of convergence. The definition of continuity at a point is given through limits. The above definition of a limit is...

problems when |t| is larger than the radius of convergence of this power series, which is given by the smallest value of |t| such that the root αj becomes...

used for determining the radius of convergence of a power series with the root test. The nth roots of 1 are called roots of unity and play a fundamental...

function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the...

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity...

definition is formal in that the series does not always have a positive radius of convergence. The Artin–Mazur zeta function is invariant under topological conjugation...

series converges absolutely at least for all complex numbers z {\displaystyle z} with | z | < 1 {\displaystyle |z|<1} ; the radius of convergence being...