Search results
Results from the WOW.Com Content Network
2 Proof. 3 Example. 4 One-sided version. 5 Example. 6 Converse of the one-sided comparison test. 7 Example. 8 See also. ... In mathematics, the limit comparison test ...
exists there are three possibilities: if L > 1 the series converges (this includes the case L = ∞) if L < 1 the series diverges. and if L = 1 the test is inconclusive. An alternative formulation of this test is as follows. Let { an } be a series of real numbers. Then if b > 1 and K (a natural number) exist such that.
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 squeeze theorem is used in calculus and mathematical analysis, typically to confirm the limit of a function via comparison with two other functions whose limits are known. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Carl Friedrich Gauss.
Advanced. Specialized. Miscellanea. v. t. e. 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.
The proof basically uses the comparison test, comparing the term f(n) with the integral of f over the intervals [n − 1, n) and [n, n + 1), respectively. The monotonic function is continuous almost everywhere.
t. e. 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 an 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.
Convergence proof techniques. Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type ...