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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 . Ratio test
Many authors do not name this test or give it a shorter name. [2] When testing if a series converges or diverges, this test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-Archimedean ultrametric triangle inequality.
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel , who proved it in 1826. [ 1 ]
This is known as the squeeze theorem. [ 1 ] [ 2 ] This applies even in the cases that f ( x ) and g ( x ) take on different values at c , or are discontinuous at c . Polynomials and functions of the form x a
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.
Summand limit (term test) Ratio; Root; Integral; ... Mean-value forms of the remainder ... the derivatives of f are known to be bounded over (a ...
A general limit can be defined if the limits to a point along all possible paths converge to the same value, i.e. we say for a function : that the limit of to some point is L, if and only if lim t → t 0 f ( s ( t ) ) = L {\displaystyle \lim _{t\to t_{0}}f(s(t))=L}
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only ...