Converging

Converging describes the process or state of approaching a specific value or condition as some parameter, often an index or time, increases.

In sequences, a_n converges to L if for every ε>0 there is N with |a_n - L| < ε for all n ≥ N. If such L exists, the sequence converges; otherwise it diverges. For functions, f_n converge pointwise to f on a domain D when, for every x in D, f_n(x) → f(x). Uniform convergence strengthens this by requiring sup_{x in D} |f_n(x) - f(x)| → 0.

Series: The partial sums S_n = ∑_{k=1}^n a_k converge to a finite limit S if S_n → S. If not, the series diverges. Criteria such as ratio or comparison tests help decide convergence; improper integrals converge when the limit of definite integrals on expanding intervals exists.

In probability, convergence of random variables is studied in several senses: almost surely, in probability, in L^p, and in distribution. These notions describe how a sequence of variables approaches a limiting variable or distribution, with implications for laws of large numbers and central limit theorems.

In numerical analysis, iterative methods generate sequences intended to approach a solution. Convergence is guaranteed under conditions like contraction, smoothness near a fixed point, or nonzero derivatives bounded away from zero. Examples include fixed-point iteration and Newton’s method, which often exhibit fast convergence near a root.

Convergence is a central concept across mathematics and related fields, defined relative to a limit object and a chosen notion of closeness or topology.