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If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]
Learn how to test infinite series for convergence using Cauchy's criterion, which relies on bounding sums of terms. Find the statement, explanation, proof and generalization of this method for complete metric spaces.
The well-known Magic Eye books feature another type of autostereogram called a random-dot autostereogram (see § Random-dot, below), similar to the first example, above. In this type of autostereogram, every pixel in the image is computed from a pattern strip and a depth map. A hidden 3D scene emerges when the image is viewed with the correct ...
Dirichlet's test is a method of testing for the convergence of a series of complex numbers. It states that if a sequence of real numbers is monotonic and bounded, and a sequence of complex numbers is non-negative and decreasing, then the series converges.
The ratio test is a criterion for the convergence of a series where each term is a nonzero real or complex number. It compares the ratio of consecutive terms and their limits, and may be extended to handle cases where the limit is 1 or fails to exist.
Here the series definitely converges for a > 1, and diverges for a < 1. When a = 1, the condensation transformation gives the series (). The logarithms "shift to the left". So when a = 1, we have convergence for b > 1, divergence for b < 1. When b = 1 the value of c enters.
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]
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.