The behaviour of a sequence, series, iterative method, or algorithm that fails to approach a finite limit or the intended solution; divergence may take the form of unbounded growth, persistent oscillation without settling, or convergence to a wrong attractor.

Principle

Convergence requires that elements form a Cauchy sequence (or satisfy contraction/monotonicity conditions in iterative schemes) in the chosen norm or topology; failure of those conditions, poor stepsize choice, incorrect fixed-point mapping or unstable discretization can produce divergence.

Demonstration

Series example: the harmonic series sum_{n} 1/n diverges (partial sums grow without bound). Iterative example: fixed-point iteration x_{k+1}=g(x_k) with |g'(x*)|>1 at the fixed point leads to divergence. Optimization: gradient descent with step size too large can diverge to infinity.

Misapplication

Labeling very slow convergence as divergence, or interpreting transient growth or cycling before asymptotic convergence as divergence; failing to check whether observed nonconvergence is due to modelling error, discretization or stopping rules.

Consequence

Diagnose cause (lack of contraction, step-size, ill-conditioning, discretization error), adjust method (smaller step sizes, damping, preconditioning), change algorithm (use alternative solvers or globalization strategies), or reformulate the problem to restore convergence.

Reversal

Convergence: the sequence or method approaches a well-defined limit or solution within the specified tolerance and topology, often guaranteed by contraction, monotone operator theory, or coercivity.

Boundary

Divergence is distinct from conditional convergence or nonabsolute convergence (series that converge conditionally are not divergent) and depends on the topology and norm; behavior under discretization or in finite precision may differ from the theoretical infinite-precision, infinite-dimensional setting.

Semantic Tension

Tension exists between divergence as mathematical nonconvergence and practical failure modes (numerical instability, ill-conditioning, algorithmic stagnation); diagnosing which phenomenon is present is critical because remedies differ.

Synthesis

Divergence denotes a failure to approach the intended limit—manifest as unbounded growth, sustained oscillation, or convergence to an unintended state—caused by violations of contraction, poor parametrization or instability; resolving divergence requires diagnosing its source and applying suitable method, parameter or model changes.