Konvergeeruda

Konvergeeruda is the Estonian verb meaning to converge, used to describe processes that move toward a common value or limit. In mathematics and related disciplines, the concept of convergence underpins how sequences, series, functions, and iterative procedures behave as they progress.

In sequences, a_n converges to L if, for every small distance ε > 0, there exists an index N such that |a_n − L| < ε for all n ≥ N. In series, the sequence of partial sums S_N = ∑_{n=1}^N a_n converges to a limit S when the partial sums get arbitrarily close to S as N increases. For functions, convergence can be pointwise, where f_n(x) → f(x) for each x in the domain, or uniform, where the convergence occurs uniformly across the domain. Convergence is contrasted with divergence, where values fail to approach a finite limit.

Beyond pure theory, konvergeeruda describes the behavior of many numerical methods and dynamical systems. Iterative algorithms converge toward a fixed point or a steady state when certain conditions are met, and convergence criteria guide stopping decisions and error estimates.

Common examples illustrate the idea: the sequence 1/n converges to 0, and the geometric series with ratio |r| < 1 converges to a/(1 − r) if it starts with a nonzero first term. If |r| ≥ 1 or if terms do not tend toward a single value, the sequence or series diverges.

See also: limit, convergence, divergence, pointwise convergence, uniform convergence, fixed-point iteration.