##  [Convergence](/convergence-0) 

 Definition

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

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

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

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

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

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

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

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

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.