Limit inferior and limit superiorIn mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Ratio testIn 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. The usual form of the test makes use of the limit The ratio test states that: if L < 1 then the series converges absolutely; if L > 1 then the series diverges; if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
Limit of a functionAlthough the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.
Direct comparison testIn 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 the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
Alternating seriesIn mathematics, an alternating series is an infinite series of the form or with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. The geometric series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not.
