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In mathematics, the nth-term test for divergence [1] is a simple test for the divergence of an infinite series:. If or if the limit does not exist, then = diverges.. Many authors do not name this test or give it a shorter name.
Probably the most interesting part of this theorem is that the Cauchy condition implies the existence of the limit: this is indeed related to the completeness of the real line. The Cauchy criterion can be generalized to a variety of situations, which can all be loosely summarized as "a vanishing oscillation condition is equivalent to convergence".
where denotes the limit superior (possibly ; if the limit exists it is the same value). If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
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. ∑ 1 / n 2 {\displaystyle \textstyle \sum 1/{n^{2}}} , and there are others for which C ...
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.
This is a list of limits for common functions such as elementary functions. In this article, the terms a , b and c are constants with respect to x . Limits for general functions
Cauchy's limit theorem, named after the French mathematician Augustin-Louis Cauchy, describes a property of converging sequences.It states that for a converging sequence the sequence of the arithmetic means of its first members converges against the same limit as the original sequence, that is () with implies (+ +) / .
Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits. [14] There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests.