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Radius of convergence. In 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 a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal ...
Convergent on the closure of the disc of convergence but not continuous sum: Sierpiński gave an example [3] of a power series with radius of convergence , convergent at all points with | | =, but the sum is an unbounded function and, in particular, discontinuous.
The radius of this disc is known as the radius of convergence, and can in principle be determined from the asymptotics of the coefficients a n. The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.
Geometrically, the two Laurent series may have non-overlapping annuli of convergence. Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at , and inner radius of convergence 0, so they both converge on an overlapping annulus.
hide. In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, [ 1 ] but remained relatively unknown until Hadamard rediscovered it. [ 2 ]
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, ) approaches from below, even in cases where the radius of convergence, of the power series is equal to and we cannot be sure whether the limit should be finite or not. See for example, the binomial series.
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It depends on the quantity | |, where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.
The existence of the radius of convergence results from the similar existence for a power series, applied to /, considered as a power series in /. It is a part of Newton–Puiseux theorem that the provided Puiseux series have a positive radius of convergence, and thus define a ( multivalued ) analytic function in some neighborhood of zero (zero ...