Convergence means approaching a definite value, as time goes on; or approaching a definite point, or a common view or opinion, or a fixed state of affairs.
A common sense example of convergence is in bargaining a price in an informal market. For example, a seller and a buyer may successively make the following offers:
- Buyer: I'll pay $10.
- Seller: Impossible! The real value is $100. How about $60?
- Buyer: Nothing more than $20. That's my final offer.
- Seller: You're really pushing me. I can't go below $40.
- Buyer: I'd rather buy in another shop. But if you would accept $30...
- Seller: OK, $30, it's a deal.
- Buyer: OK.
Here the sequence of bids and counter-bids evidently converges, quite rapidly, to a common price.
More formally, 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 a limit, which may be itself unknown. For any fixed standard of accuracy, you can always be sure to be within it, provided you have gone far enough.
The opposite of convergence is divergence. It may be some kind of oscillation, unrestricted growth (recognised as the case of an infinite limit), or chaotic behavior. In particular cases the definitions of 'far enough' metric and the other terms vary.
An infinite series that is divergent does not a priori have any mathematical value. 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 series (mathematics) for a longer discussion including history and various types of convergence.
See also
Other meanings