Property of a sequence, series, or family of approximations whereby its members approach a single limiting value (often the exact solution) as the refinement parameter (e.g., iteration count, mesh size, or truncation index) tends to its ideal limit.

Principle

An algorithm or sequence converges when the distance between its nth approximation and the limit tends to zero as n increases or as the discretization parameter tends to its ideal value; the rate at which this distance shrinks characterizes the convergence speed.

Demonstration

Iterating Newton's method on a sufficiently smooth nonlinear equation: close to a simple root, successive iterates typically approach the true root and the error often decreases quadratically, illustrating convergence and a specific rate of convergence.

Misapplication

Treating observed movement of iterates toward a stable-looking value on a coarse test as proof of convergence for all initial data or finer refinements; mislabeling slow numerical stagnation or oscillation for convergence.

Consequence

When convergence holds and its rate is known, one can predict how many iterations or how fine a discretization is required to reach a target tolerance and allocate computational resources accordingly.

Reversal

Divergence: the sequence fails to approach any finite limit, or oscillates without settling, or the error does not tend to zero—often a sign of an unsuitable method, improper parameters, or instability.

Boundary

Applies to deterministic sequences, series, and iterative schemes; does not by itself quantify absolute correctness on finite steps, nor does it imply stability or accuracy without additional conditions; probabilistic estimators require separate modes of convergence.

Semantic Tension

Convergence is often conflated with accuracy or stability: accuracy speaks to closeness on finite samples, stability to response to perturbations, while convergence is an asymptotic statement about limits.

Synthesis

Convergence is the asymptotic approach of approximations to a limit; its practical value comes from knowing whether and how fast that approach happens so that finite computations can be planned and interpreted, always checked against stability and accuracy considerations.