Convergence is the approach of a sequence, series, net, filter, or other limiting process to a limit in a specified sense. Its principal meanings depend on the structure under consideration, such as a topology, metric, norm, measure, or probability measure.
In a topological space, a net converges to a point 
if it is eventually in every neighborhood
of 
, while a filter converges
to 
if every neighborhood
of 
belongs to the filter.
The usual definition of a convergent sequence
in a metric space is the corresponding special case
for a sequence. A series converges when its sequence
of partial sums converges.
Absolute convergence and conditional
convergence distinguish whether the associated series
of absolute values also converges.
For a sequence of functions, common modes include pointwise convergence,
uniform convergence, and almost
everywhere convergence. On a measure space 
, the functions 
exhibit convergence
in measure to 
if, for every 
,
 |
In a normed space, norm convergence is also called strong convergence, while weak convergence tests convergence using linear functionals.
For random variables, important modes are convergence almost surely, convergence
in probability, convergence in distribution,
and convergence in mean. Convergence almost
surely implies convergence in probability,
which implies convergence in distribution.
Convergence in the rth mean for 
also implies convergence
in probability.
In numerical algorithms, the rate of convergence describes how quickly approximations approach a limit, while convergence improvement accelerates this approach. Convergence tests determine whether a series converges. The opposite of convergence is divergence.
