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...

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

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...

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...

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...

convergence – Domain of convergence of power series Riemann series theorem – Unconditional 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...

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...

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} ...

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

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...

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...

} so the order of convergence of the geometric series is 1 and its rate of convergence is | r | {\displaystyle |r|} . Convergence of order one is also...

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

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,...

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...

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...

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...

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 →...

capital Radius of curvature (optics) Receiver operating characteristic, ROC curve (statistics) Radius of convergence Rail operating centre, a type of railway...

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

renormalizable theories. An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero. The basic argument goes...

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...

only if n is not a power of 2. It follows from Legendre's formula that the p-adic exponential function has radius of convergence p − 1 / ( p − 1 ) {\displaystyle...

related convergence test, also known as Abel's test, can often be used to establish the convergence of a power series on the boundary of its circle of convergence...