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. It is particularly useful in connection with power series.
The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test. For a series
the root test uses the number
where "lim sup" denotes the limit superior, possibly +∞. Note that if
converges then it equals C and may be used in the root test instead.
The root test states that:
if C < 1 then the series converges absolutely,
if C > 1 then the series diverges,
if C = 1 and the limit approaches strictly from above then the series diverges,
otherwise the test is inconclusive (the series may diverge, converge absolutely or converge conditionally).
There are some series for which C = 1 and the series converges, e.g. , and there are others for which C = 1 and the series diverges, e.g. .
This test can be used with a power series
where the coefficients cn, and the center p are complex numbers and the argument z is a complex variable.
The terms of this series would then be given by an = cn(z − p)n. One then applies the root test to the an as above. Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately). A corollary of the root test applied to such a power series is the Cauchy–Hadamard theorem: the radius of convergence is exactly taking care that we really mean ∞ if the denominator is 0.
The proof of the convergence of a series Σan is an application of the comparison test.
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In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
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 an 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. The usual form of the test makes use of the limit The ratio test states that: if L < 1 then the series converges absolutely; if L > 1 then the series diverges; if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series . If the limit of the summand is undefined or nonzero, that is , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.
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