Convergence

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.

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:

Related Topics:
Mathematics - Sequence - Series - Limit

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• 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 the speed of convergence that 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 pertain 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 (no clear introductory statement of context).
• 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.

Related Topics:
Divergence - Oscillation - Chaotic - Generating function - Asymptotic series - Summation method

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