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...

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

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...

_{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...

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...

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...

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...

ratio test for convergence and in the definitions of rates of convergence. Geometric series have further served as prototypes in the study of mathematical...

}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...

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

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

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...

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} ...

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 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...

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...

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...

{d} M} goes to infinity at these values of M {\displaystyle M} . This means that the radius of convergence of the Maclaurin series is cosh − 1 ⁡ ( 1 /...

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

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 → ∞...

it is possible to show that this power series has an infinite radius of convergence and so defines ez for all complex z. For complex z e z = lim n →...

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...

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...

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...

of convergence centered at the point c in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this...

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...

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...

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

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...