Convergence

For other uses, see Convergence (disambiguation).

In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. Convergent can denote a rational approximation, or can be used as an adjective.

Mathematics

Main article: Modes of convergence

In mathematics, convergence describes limiting behaviour, particularly of an infinite sequence or series toward some limit. To assert convergence is to claim the existence of such a limit, which may be itself unknown. For any fixed standard of accuracy, however, you can always be sure to be within that limit, provided you have gone far enough. The following lists more specific usages of this word:

• Convergent sequence, a sequence which has a limit, see limit of a sequence.
• Convergent series, a sequence of which the partial sums have a limit.
• Convergent of a (possibly infinite) continued fraction.
• Convergent net - a generalization of a convergent sequence.
• Convergent filter - a generalization of a convergent net.
• Integral test for convergence is a technique used to test infinite series of nonnegative terms for convergence.
• Radius of convergence pertains to a domain interval over which a power series converges.
• Uniform convergence pertains to pointwise convergence where the speed of convergence is independent of any value in the domain.
• Monotone convergence theorem pertains to any one of several such theorems defined over a monotone sequence of numbers.
• Convergence of random variables pertains to any one of several notions of convergence in probability theory.
• Rate of convergence pertains to the "speed" at which a convergent sequence approaches its limit.
• Absolute convergence pertains to whether the absolute value of the limit of a series or integral is finite.
• Pointwise convergence is the convergence of functions' values at each specific input individually.
• Gromov-Hausdorff convergence pertains to metric spaces and is a generalization of Hausdorff distance.
• Convergence of Fourier series pertains to whether the Fourier series of a periodic function converges. Also known as classic harmonic analysis.
• Dominated convergence theorem pertains to a theorem by Henri Lebesgue.
• Convergence of a numerical method for solving differential equations.

The opposite of convergence is divergence. Divergence may be some kind of oscillation, unrestricted growth (recognised as the case of an infinite limit), or chaotic behavior. An infinite series that is divergent cannot be used for meaningful computations of its value. Nevertheless, divergent series can be summed formally, as generating functions or asymptotic series, or via some summation method.

Natural sciences

• Convergent evolution pertains to organisms not closely related that independently acquire similar characteristics while evolving in separate and sometimes varying ecosystems.
• Convergent synthesis is a strategy that aims to improve the efficiency of multi-step chemical synthesis.
• Convergent boundary is a fault boundary defined in the specialty of geology known as plate tectonics.
• Convergence zone in meteorology is an area where the horizontal wind produces a net in-flow of air
  • South Pacific convergence zone is a belt of low pressure from the west Pacific warm pool
  • Intertropical convergence zone is a belt of low pressure at the equator.
• Convergence is the simultaneous inward movement of both eyes toward each other, usually in an effort to maintain single binocular vision when viewing an object.

Computing and technology

• Convergence (evolutionary computing) is a means of modelling the tendency for genetic characteristics of populations to stabilize over time.
• Premature convergence is an anomaly in Evolutionary computation in which the population evolved to some stable yet sub-optimal state.
• Technological convergence refers to a trend where some technologies having distinct functionalities evolve to technologies that overlap, i.e. multiple products come together to form one product, with the advantages of each initial component.

Social sciences

• Language convergence pertains to the blending of two languages that are perceived as having equal social status. Opposite of Non-convergent discourse.
• Non-convergent discourse pertains to the persistence of asymmetric or bilingual discourse in natural languages.
• Catch-up effect is otherwise known as the Theory of convergence in economic theory.
• Convergence criteria are requirements specified by the European Union that stipulate the membership qualifications each state must fulfill.
• Media convergence refers to the removal of entry barriers across the IT, telecoms, media and consumer electronics industries, creating one large 'converged' industry. In recent years it has also come to refer specifically to the ongoing 'bundling' of services by telecoms players, resulting in 'triple play' (one company offering combined fixed line phone, TV and broadband services) and more recently 'quad play' (triple play plus mobile phone). The 2006 acquisition of Virgin Mobile by NTL is often cited as an exemplar of the move in this direction.

See also

• Divergence (disambiguation)