Convergence

In mathematics, convergence describes limiting behaviour, particularly of an infinite sequence or series. To assert convergence is to claim the existence of a limit, such that by going far enough the limiting value is approximated, and never subsequently is the approximation worse. In particular cases the definitions of 'far enough' and the other terms vary.

The opposite of convergence is divergence, which may be some kind of oscillation of values, or unrestricted growth (recognised as the case of an infinite limit). An infinite series that is divergent does not a priori have any mathematical content. That is, it cannot be used for meaningful computations of its value. Such series are indeed applied: as generating functions, as asymptotic series, or via some summation method.

In general, 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.

See also net (topology), uniform convergence.

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