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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]
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series. Integral test. The series can be compared to an integral to establish convergence or divergence. Let () = be a positive and monotonically decreasing function. If
In mathematics, convergence tests are methods to determine if an infinite series converges or diverges. Pages in category "Convergence tests" The following 17 pages are in this category, out of 17 total.
Download as PDF; Printable version; ... Convergence tests (17 P) D. Divergent series (2 C, 15 P) L. ... Convergence problem; Convergent series;
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analyse 1821.
A famous example of an application of this test is the alternating harmonic series = + = + +, which is convergent per the alternating series test (and its sum is equal to ), though the series formed by taking the absolute value of each term is the ordinary harmonic series, which is divergent.
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.