• 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...
    16 KB (2,616 words) - 00:14, 9 September 2024
  • 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...
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  • Thumbnail for Radius
    plane. Bend radius Filling radius in Riemannian geometry Mean radius Radius of convergence Radius of convexity Radius of curvature Radius of gyration Semidiameter...
    10 KB (1,190 words) - 04:09, 12 November 2024
  • 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...
    6 KB (1,145 words) - 17:22, 10 December 2024
  • _{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...
    20 KB (3,886 words) - 19:24, 31 October 2024
  • Thumbnail for Taylor series
    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...
    48 KB (8,253 words) - 14:55, 12 December 2024
  • 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...
    2 KB (314 words) - 06:41, 13 March 2024
  • 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...
    8 KB (1,538 words) - 03:54, 7 August 2024
  • Domain of convergence of power series Riemann series theorem – Unconditionally convergent series converge absolutely Unconditional convergence – Order-independent...
    28 KB (5,194 words) - 17:22, 3 November 2024
  • Thumbnail for Laurent series
    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...
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  • Thumbnail for Analyticity of holomorphic functions
    }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...
    6 KB (1,136 words) - 23:43, 16 May 2023
  • 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} ...
    2 KB (283 words) - 00:21, 20 November 2023
  • 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...
    7 KB (946 words) - 10:15, 22 December 2024
  • whenever α {\displaystyle \alpha } is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (5), by comparison...
    14 KB (1,904 words) - 02:40, 21 December 2024
  • 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,...
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  • Thumbnail for Extrapolation
    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...
    14 KB (1,876 words) - 20:02, 1 December 2024
  • 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...
    7 KB (1,013 words) - 04:07, 26 November 2024
  • notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The...
    41 KB (5,280 words) - 21:51, 25 September 2024
  • Thumbnail for Analytic function
    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...
    15 KB (2,178 words) - 12:45, 30 November 2024
  • series converges absolutely at least for all complex numbers z {\displaystyle z} with | z | < 1 {\displaystyle |z|<1} ; the radius of convergence being...
    10 KB (1,762 words) - 09:30, 19 November 2024
  • divergent. Convergence means there is a value after summing infinitely many terms, whereas divergence means no value after summing. The convergence of a geometric...
    33 KB (4,734 words) - 02:50, 16 December 2024
  • 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...
    13 KB (2,439 words) - 14:00, 8 November 2024
  • 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...
    10 KB (1,999 words) - 18:39, 27 September 2023
  • Root test (category Convergence tests)
    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 → ∞...
    10 KB (1,926 words) - 18:15, 12 August 2024
  • Thumbnail for Wilkinson's polynomial
    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...
    14 KB (2,171 words) - 01:38, 23 September 2024
  • Return of capital Return on capital Radius of curvature (optics) Receiver operating characteristic, ROC curve (statistics) Radius of convergence Rail operating...
    5 KB (672 words) - 20:05, 10 November 2024
  • Thumbnail for Euler's formula
    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 →...
    26 KB (3,851 words) - 17:07, 26 November 2024
  • 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...
    32 KB (4,767 words) - 14:23, 9 December 2024
  • 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...
    36 KB (5,973 words) - 03:41, 6 December 2024
  • 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...
    87 KB (14,363 words) - 13:47, 4 November 2024