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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 ...
If = and = + (), then both series have the same radius of convergence of 1, but the series = (+) = = has a radius of convergence of 3. The sum of two power series will have, at minimum, a radius of convergence of the smaller of the two radii of convergence of the two series (and it may be higher than either, as seen in the example above).
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.
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix. This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C). In this case, for the graph G define:
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 ]
Entire function. In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions ...
Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately).
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.