konvergiert

Konvergiert is the third-person singular present form of the German verb konvergieren and is used in mathematics to describe the tendency of a sequence, function, or series to approach a limit as an index or parameter grows.

In the context of sequences, a sequence (a_n) converges to a limit L if there exists a real number L such that for every ε > 0 there exists N with |a_n − L| < ε for all n ≥ N. If such an L exists, the sequence is called convergent; otherwise it is divergent. The limit L may also be defined in the extended real numbers.

For functions, one speaks of pointwise convergence: a sequence of functions f_n defined on a common domain D converges pointwise to f if, for every x in D, lim_{n→∞} f_n(x) = f(x). A stronger notion is uniform convergence, which requires that sup_{x∈D} |f_n(x) − f(x)| → 0 as n → ∞; uniform convergence preserves many properties, such as continuity under suitable conditions.

In the context of series, konvergiert refers to the convergence of the sum ∑_{n=1}^∞ a_n. The series converges if the sequence of partial sums S_N = ∑_{n=1}^N a_n converges to a limit. Absolute convergence (∑ |a_n| converges) implies convergence, while conditional convergence may occur without absolute convergence. Common tests include the ratio test, root test, and comparison tests.

Common examples include the sequence a_n = 1/n, which converges to 0; the harmonic series ∑ 1/n, which diverges; and the geometric series ∑ r^n, which converges to 1/(1−r) for |r| < 1. Konvergenz is central in analysis, underpinning limits, continuity, and integration.