differansegrense

Differansegrense, a term used in Norwegian mathematics, refers to the limit of the difference between successive terms of a sequence, if such a limit exists. For a real sequence (a_n), the differansegrense is defined as L = lim_{n→∞} (a_{n+1} − a_n), provided the limit exists. The value L describes the asymptotic linear growth rate of the sequence. If L exists and is finite, then a_n grows roughly like L n for large n, in the sense that a_n = L n + o(n).

Example: If a_n = 3n + 7, then a_{n+1} − a_n = 3 for all n, so the differansegrense is 3. Conversely, if a_n = n^2, then a_{n+1} − a_n = 2n + 1, which grows without bound, so there is no finite differansegrense.

Relation to other concepts: For a function f defined on the integers, the discrete difference Δf(n) = f(n+1) − f(n) is the differansegrense of the sequence f(n). If f is a smooth function and n is large, Δf(n) often behaves like f′(n). In numerical analysis, finite-difference methods use differences to approximate derivatives, connecting differansegrense to notions of rates of change and linear growth.

In practice, the differansegrense helps characterize long-term behavior of sequences and is a basic tool in asymptotic analysis and recurrence relations. See also finite difference, difference equation, and limits of sequences.