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Calculus ′ = () ... is a strictly monotone and divergent sequence and the following limit exists: ... is finitely convergent if its ratio is less than one ...
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 whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
In calculus, the Abel–Dini–Pringsheim theorem is a convergence test which constructs from a divergent series a series that diverges more slowly, and from convergent series one that converges more slowly. [1]: §IX.39 Consequently, for every convergence test based on a particular series there is a series about which the test is inconclusive.
for the infinite series. Note that if the function () is increasing, then the function () is decreasing and the above theorem applies.. Many textbooks require the function to be positive, [1] [2] [3] but this condition is not really necessary, since when is negative and decreasing both = and () diverge.
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.
A series is convergent (or converges) if and only if the sequence (,,, … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit ; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices , one gets partial sums that become closer and closer to a given number.
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. [1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
If such a limit exists and is finite, the sequence is called convergent. [2] A sequence that does not converge is said to be divergent . [ 3 ] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
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