Cauchy–Hadamard theorem

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, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis. Consider the formal power series in one complex variable z of the form where Then the radius of convergence of f at the point a is given by where lim sup denotes the limit superior, the limit as n approaches infinity of the supremum of the sequence values after the nth position. If the sequence values are unbounded so that the lim sup is ∞, then the power series does not converge near a, while if the lim sup is 0 then the radius of convergence is ∞, meaning that the series converges on the entire plane. Without loss of generality assume that . We will show first that the power series converges for , and then that it diverges for . First suppose . Let not be or For any , there exists only a finite number of such that . Now for all but a finite number of , so the series converges if . This proves the first part. Conversely, for , for infinitely many , so if , we see that the series cannot converge because its nth term does not tend to 0.