convergence

convergence

[kuhn-vur-juhns]

Mathematical property of infinite series, integrals on unbounded regions, and certain sequences of numbers. An infinite series is convergent if the sum of its terms is finite. The series 12 + 14 + 18 + 116 + 132 + ... sums to 1 and thus is convergent. The harmonic series 1 + 12 + 13 + 14 + 15 + ... does not converge. An integral calculated over an interval of infinite width, called an improper integral, describes a region that is unbounded in at least one direction. If such an integral converges, the unbounded region it describes has finite area. A sequence of numbers converges to a particular number when the difference between successive terms becomes arbitrarily small. The sequence 0.9, 0.99, 0.999, etc., converges to 1.

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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 is the adjectival form, and also a noun meaning an iterative approximation.

Mathematics

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:

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
  • 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

  • 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.
  • Convergence (telecommunications) refers to the combination of multiple services through lines of telecommunication from a single provider.
  • Convergence (evolutionary computing) is a means of modeling 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.

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.
  • Telecommunication convergence is a marketing concept with heavy technological implications, referring to the bundling of services by telecoms players, both in terms of the product portfolio they offer (vertical convergence), and of the channels through which their products are sold and serviced (horizontal convergence).
  • Futures and cash market price convergence describes the tendency of the futures price and cash price at the delivery location to come together as the delivery period approaches. When in the delivery period, at the delivery location, convergence will have occurred when the futures price and cash price are economically equivalent. This is also used as GATE icons. (GATE Classes)

See also

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