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In mathematics, the ratio test is a test (or "criterion") for the convergence of a series =, where each term is a real or complex number and a n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.
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.
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]
If r = 1, the root test is inconclusive, and the series may converge or diverge. The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse ...
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
For instance, ideally the solution of a differential equation discretized via a regular grid will converge to the solution of the continuous equation as the grid spacing goes to zero, and if so the asymptotic rate and order of that convergence are important properties of the gridding method.
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.
Suppose f is analytic in a neighborhood of a and f(a) = 0.Then f has a Taylor series at a and its constant term is zero. Because this constant term is zero, the function f(x) / (x − a) will have a Taylor series at a and, when f ′ (a) ≠ 0, its constant term will not be zero.