Radius of convergence information

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...

that plane. Bend radius Filling radius in Riemannian geometry 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...

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...

_{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...

}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...

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} ...

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...

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...

number r, called the radius of convergence such that the series converges if T is substituted for a nonzero complex number t of absolute value less than...

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 → ∞...

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

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