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Calculus. 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.
Rational root theorem. In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer coefficients and . Solutions of the equation are also called roots or zeros of the polynomial on the left side.
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.
Unit root test. In statistics, a unit root test tests whether a time series variable is non-stationary and possesses a unit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either stationarity, trend stationarity or explosive root depending on the test used.
In statistics, the Phillips–Perron test (named after Peter C. B. Phillips and Pierre Perron) is a unit root test. [1] That is, it is used in time series analysis to test the null hypothesis that a time series is integrated of order 1. It builds on the Dickey–Fuller test of the null hypothesis in , where is the first difference operator.
Augmented Dickey–Fuller test. In statistics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis depends on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller test ...
The root test shows that its radius of convergence is 1. In accordance with this, the function f ( z ) has singularities at ± i , which are at a distance 1 from 0. For a proof of this theorem, see analyticity of holomorphic functions .
The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space. If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series.