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. In mathematics, the point is an equilibrium point for the differential equation if for all . ...

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: In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...

• Convergent series provides a general mathematical definition and a context in which to understand the remaining mathematical usages.
• In topology, an infinite sequence of points of a topological space is said to converge to a point x if every neighborhood of x contains all but a finite number of points of the sequence.
• 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

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. In mathematics, a series is the sum of the terms of a sequence of numbers. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... Monotone convergence theorem, in mathematics, may refer to several theorems, all of which are concerned with a monotonic function in one way or another: Monotonic function refers to the convergence of an infinite series that is monotonic Dominated convergence theorem refers to Lebesgues monotone convergence theorem Categories: | | ... In probability theory, there exist several different notions of convergence of random variables. ... In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. ... In mathematics, a series is a sum of a sequence of terms. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ... Gromov-Hausdorff convergence is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. ... In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called classic harmonic analysis, a branch of pure mathematics. ... In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called classic harmonic analysis. ... In mathematics, Lebesgues dominated convergence theorem states that if a sequence { fn : n = 1, 2, 3, ... } of real-valued measurable functions on a measure space S converges almost everywhere, and is dominated (explained below) by some nonnegative function g in , then To say that the sequence is dominated by... In mathematics, a divergent series is an infinite series that does not converge. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... In mathematics, a divergent series is an infinite series that does not converge. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... In mathematics, a divergent series is a series that does not converge. ...

Natural sciences

• Convergent evolution pertains to organisms not closely related that independently acquire similar characteristics while evolving in separate and sometimes varying ecosystems.
• Convergent synthesisis 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.

In evolutionary biology, convergent evolution is the process whereby organisms not closely related, independently evolve similar traits as a result of having to adapt to similar environments or ecological niches. ... In chemistry a convergent synthesis is a strategy that aims to improve the effeciency of multi-step chemical synthesis. ... In plate tectonics, a convergent boundary (convergent fault boundary, convergent plate boundary, or active margin) is where two tectonic plates slide towards each other and usually collide forming either a subduction zone with its associated island arc or an orogenic belt and associated mountain range. ... Convergence zone usually refers to a region in the atmosphere where two prevailing flows meet and interact, usually resulting in distinctive weather conditions. ... The South Pacific Convergence Zone (SPCZ) is a band of low-level convergence, cloudiness and precipitation extending from the west Pacific warm pool south-eastwards towards French Polynesia. ... The thunderstorms of the Intertropical Convergence Zone form a line across the eastern Pacific Ocean. ... In ophthalmology, 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 ^ . It is a type of vergence eye movement. ...

Computing and technology

• Convergence (converged environments/networks) defines a multi-media environment and/or network where signals regardless of type (i.e. voice, quality audio, video, data, etc.) and encoding methodology may be seamlessly exchanged between independent endpoints with similar characteristics. Convergence in this case requires the overall environment have two primary characteristics: 1) the intelligence to provide translation between disparent signal types and multipoint routing to establish connectivity between requested endpoints 2) The ability to dynamically allocate required bandwidth to support endpoint requirements for each requested session. Convergence as defined by this is independent of signal format and transport media.

Convergence (converged environments/networks), as such, was formally defined in January 1992 by Phillip A. Coombs, Manager of Systems Integration for McGraw Broadcast Communications and President of the Gemini Group, Cincinnati, Ohio. This environment is fully described in US Patent 5,577,042.

• Convergence and "convergence time" are a process and a measure, respectively, of the adaptation of a computer network to unplanned changes in its topology or structure. For example, a routing protocol's convergence time is how long it takes between when a link is broken to when all of the routers (nodes) in the network have restructured their routing tables to take the next most optimal path.
• 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 each initial component.

Routing protocols allow different computer networks to communicate. ... Precisely every individual in the population is identical. ... This loosely means that something has gone wrong. ... Technological convergence is the modern presence of a vast array of different types of technology to perform very similar tasks. ...

Telecommunications

In telecommunications convergence takes on a variety of definitions. The four most common include industry convergence, network convergence, platform convergence and application convergence.

Industry convergence is taking place across traditional business segments creating new competitors and markets. Four business industries are at the confluence of industry convergence including: electronics, media, IT and Telecom. The borders between these industries are being blurred through mergers, acquisitions, partnerships and bundled product offerings such as the triple play and quadruple play. The bundling of products and services into one offering has obvious advantages to consumers but the most significant benefit, reduced churn rates, goes to the providers. High churn rates have a significant negative impact on the profitability of telecommunications companies and bundling has reduced churn rates dramatically.

Network convergence is most notably taking place with the merging of disparate voice and data networks onto one integrated network that is access and information agnostic. The motivation for network convergence is twofold: 1) cost containment through the integration of multiple networks operating with disparate order-entry and billing systems onto one network and 2) improved product offerings made capable because IP networks can be updated faster and for less money than older application specific networks.

Platform convergence is taking place on multiple levels. Two examples include: mobile phones now regularly incorporate cameras, music players and most recently video players while computers now offer tuners for viewing TV signals.

Application convergence is the merging of disparate and sometimes competing applications into one product. Sometimes referred to as Mash-ups, these new applications have the potential to revolutionize the web.

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.
• In the context of bargaining, Convergence pertains to a behavior in which the price offered by a buyer may increase while the price acceptable to a seller may decrease until both prices approach equality, in which case they are said to converge.
• 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.
• Economic convergence refers to the phenomena in which lesser developed countries supposedly catch up to developed countries in terms of economic productivity and growth. Many theorists agree that economic growth in the United States peaked in the first half of the 20th century allowing it to diverge from other nations and leapfrogging to its status as a world leader in nearly every area. However, other countries are starting to gain on the United States. One of the sources or causes of growth in the past, especially the 19th century was the accumulation of factors of production. Today, economic growth and productivity seems to be tied to innovation and technology.