In mathematics, **convergence** describes limiting behaviour, particularly of an infinite sequence or series (mathematics). 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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