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...

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...

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...

_{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...

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...

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,...

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...

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 → ∞...

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...

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} ...

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

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...

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...

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...

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...

function that cannot be analytically continued anywhere outside the radius of convergence within which it is defined by a power series. The word lacunary...

series have the same radius of convergence as the original series. Assuming that [a − r, a + r] ⊂ I and r < R, all these series converge uniformly on (a −...

where y0 is a solution of the first kind. Its radius of convergence is at least as large as the minimum of the radii of convergence of p ( x ) {\displaystyle...

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

coefficients). Technically, the generating function is scaled to have radius of convergence 1, so it has singularities on the unit circle – thus one cannot...

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 →...

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...

combination of radius of convergence for a one complex variable. This combination is generally not unique and there are an infinite number of combinations...

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...

divergent. Convergence means there is a value after summing infinitely many terms, whereas divergence means no value after summing. The convergence of a geometric...