grenscriterium

Grenscriterium, in Dutch mathematics, is a generic term for a criterion that uses a limit (grens) to decide whether a mathematical object has a given property. The term is used across areas such as sequences, series, integrals, and numerical methods, and it emphasizes the role of limiting behavior in establishing results. A grenscriterium does not specify a single theorem; rather, it describes a class of tests or conditions that depend on the existence and value of a limit.

In sequences and series, the most common instances are limit-based tests. If a sequence a_n has a limit L, then the sequence converges to L, which is itself a convergence criterion. For infinite series, limit-based tests include the ratio test, where lim |a_{n+1}/a_n| = q, and if q < 1 the series converges; and the root test, where lim sup |a_n|^{1/n} = r < 1 implies convergence. Both rely on a limiting quantity to decide convergence.

In numerical analysis and applied contexts, grenscriterium concepts describe stability and convergence of algorithms by looking at the limit of error ratios or residuals. For example, if the limit of the ratio of consecutive errors is a number less than one, the method is linearly convergent. Similar limit-based criteria are used in the analysis of iterative solvers and in the study of approximations near boundary points in differential equations.

History and usage: The term is common in Dutch textbooks and papers as a general descriptor rather than a named theorem. It overlaps with standard limit concepts in calculus, analysis and numerical mathematics.