convergence

Convergence is the idea that a process, sequence, function, or sequence of random variables approaches a limiting value as its index or input tends to a specified limit, usually infinity. It is a foundational concept across mathematics, statistics, and numerical analysis, capturing the notion of stabilization or predictability in the limit.

For a sequence {a_n} in a metric space with limit L, convergence means that the distance d(a_n, L) tends to zero as n grows. In the real numbers this implies that, beyond some index, all terms are arbitrarily close to L and, if a limit exists, it is unique.

In analysis, functions can converge in several senses. Pointwise convergence f_n(x) -> f(x) means the convergence holds for every x in the domain. Uniform convergence strengthens this by requiring sup_x |f_n(x) - f(x)| to go to zero; uniform convergence preserves many limits and integrals and allows justified interchange of limits with differentiation under suitable conditions.

Series converge when the sequence of partial sums S_N = sum_{n=1}^N a_n approaches a finite limit. Absolute convergence implies convergence of the series regardless of sign, while conditional convergence may depend on the order of terms.

In probability and measure theory, convergence concepts include convergence in probability, almost sure convergence, convergence in distribution, and L^p convergence, each with its own implications and relationships (for example, almost sure convergence implies convergence in probability).

Numerical analysis considers convergence of iterative methods to a solution, with rates such as linear or quadratic convergence describing how quickly the iterates approach the limit.

Example: a_n = 1/n converges to 0. The function f_n(x) = x^n on [0,1] converges pointwise to 0 for x in [0,1), but not uniformly on [0,1].