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The integral test applied to the harmonic series. Since the area under the curve y = 1/x for x ∈ [1, ∞) ... To see the divergence of the series ...
This is also known as the nth-term test, test for divergence, or the divergence test. ... A commonly-used corollary of the integral test is the p-series 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.
Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral , according to the integral test for convergence .
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.
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field of non-zero order k is written as =, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar.
Furthermore, if Σa n is divergent, a second divergent series Σb n can be found which diverges more slowly: i.e., it has the property that lim n->∞ (b n /a n) = 0. Convergence tests essentially use the comparison test on some particular family of a n, and fail for sequences which converge or diverge more slowly.