Nichtabnehmenden

Nichtabnehmenden is the attributive inflected form of the German adjective nichtabnehmend, used in mathematics to describe quantities that do not decrease. In standard usage, nichtabnehmend denotes non-decreasing (monotone increasing) behavior: for a sequence or function, the value does not go down as the argument increases.

In sequences, a_n is nichtabnehmend if a_{n+1} ≥ a_n for all n. This means the sequence either stays constant or increases. The opposite property is abnehmend, where a_{n+1} ≤ a_n for all n. A simple example of nichtabnehmend is 3, 3, 4, 4, 7, 7.

In the context of functions, a function f is nichtabnehmend on an interval if x ≤ y implies f(x) ≤ f(y) for all x, y in the interval. This means the function never decreases as its input grows. If f is differentiable, a nonnegative derivative on the interval is a sufficient condition for nichtabnehmend behavior, though not necessary.

Analytically, monotone nichtabnehmende sequences that are bounded above converge to their supremum (Monotone Convergence Theorem). For functions that are nichtabnehmend on a compact interval, the function attains its maximum at the right endpoint.

The form nichtabnehmenden appears in plural contexts, for example as die nichtabnehmenden Werte (the non-decreasing values). It reflects the grammatical agreement of das Adjektiv mit dem Nomen in Plural.