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:

• Modes of convergence
• 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