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In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test .
If p ≤ 0, then the nth-term test identifies the series as divergent. If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence. If 1 < p, then the nth-term test is inconclusive, but the series is convergent by the integral test for convergence.
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 mathematics, convergence tests are ... A commonly-used corollary of the integral test is the p-series test. Let > ... (M n) satisfying the conditions ...
In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f ( n ) {\displaystyle f(n)} of non-negative real numbers , the series ∑ n = 1 ∞ f ( n ) {\textstyle \sum \limits _{n=1}^{\infty }f(n)} converges if and only if the "condensed ...
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.
The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a weak form of the Euler–Lagrange equation. The theorem of Du Bois-Reymond ...
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field.