konverges

Konverges is a term used in mathematics to describe the tendency of a sequence, function, or series to approach a limiting value. It is a central idea in calculus and analysis. In sequences, a sequence (x_n) konverges to a limit L if for every epsilon > 0 there exists N such that |x_n − L| < epsilon for all n ≥ N. The limit, when it exists, is unique. In metric spaces one says a sequence (x_n) konverges to x if the distance d(x_n, x) tends to zero.

In the context of functions, konvergence can be pointwise or uniform. A sequence of functions f_n konverges pointwise to f if, for every x in the domain, f_n(x) → f(x). It konverges uniformly if sup_x |f_n(x) − f(x)| → 0 as n → ∞. For series, the partial sums s_N = sum_{n=1}^N a_n konverge to a finite limit S if s_N → S as N → ∞.

Convergence can fail in various ways; a function sequence can konverge to a limit function that is discontinuous, and a series can diverge to infinity or fail to settle. In complete metric spaces, a sequence konverges if and only if it is Cauchy; that is, for every epsilon > 0 there exists N such that d(x_n, x_m) < epsilon for all n, m ≥ N. Convergence tests and theorems provide practical criteria for establishing konvergence in different contexts.