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In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement [ edit ]
If r > 1, then the series diverges. 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. [1]
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.
The integral test applied to the harmonic series. Since the area under the curve y = 1/ x for x ∈ [1, ∞) is infinite, the total area of the rectangles must be infinite as well. Part of a series of articles about
Comparison test can mean: Limit comparison test , a method of testing for the convergence of an infinite series. Direct comparison test , a way of deducing the convergence or divergence of an infinite series or an improper integral.
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The Lévy–Steinitz theorem identifies the set of values to which a series of terms in R n can converge. A typical conditionally convergent integral is that on the non-negative real axis of (see Fresnel integral).
in which taking the limit first with respect to n gives 0, and with respect to m gives ∞. Many of the fundamental results of infinitesimal calculus also fall into this category: the symmetry of partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators.
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