talrække

Talrække is a term used in Danish and Norwegian for a mathematical sequence, i.e., an ordered list of numbers (a1, a2, a3, ...) arranged according to a rule or pattern. The elements are indexed by natural numbers; a finite talrække has a last term, while an infinite talrække continues indefinitely. Each term is typically a real or complex number, and the notation a_n is used for the nth term.

Types: A talrække can be given explicitly by a formula a_n = f(n) or implicitly by a recurrence relation that defines a_n in terms of earlier terms. Common special cases include arithmetic progression (a_n = a1 + (n-1)d) where successive terms differ by a constant d, and geometric progression (a_n = a1 · r^(n-1)) where successive terms multiply by a constant r.

Properties: Sequences can be monotonic (always increasing or decreasing); bounded (having an upper and/or lower bound); convergent, meaning a_n tends to a limit L as n → ∞. If no finite limit exists, the sequence diverges. For infinite sequences, notions such as convergence of the sequence (not the sum) are central; convergence tests relate to related sums called series, where one studies the partial sums S_N = a1 + a2 + ... + a_N.

Applications: Talrækker are used to describe patterns in numbers, model processes over time, analyze algorithms, and serve as a foundation for more advanced topics in calculus and analysis.