röðunnar

Röðunnar is the Icelandic term used to refer to a sequence in mathematical and scientific contexts. The base noun röð means sequence, row, or line, and röðunnar is its genitive form, used in phrases such as “eiginleikar röðunnar” (the properties of the sequence).

A sequence is a function from the natural numbers to a set (often the real or complex numbers). It is described by its nth term a_n, where n starts at 1 or 0. The sequence is typically written as {a_n}. Each term is determined by its position in the sequence, and the terms are indexed in order.

Common types include arithmetic sequences with constant difference d, where a_n = a_1 + (n−1)d, and geometric sequences with constant ratio r, where a_n = a_1 r^{n−1}. Convergence and divergence describe whether a_n approaches a limit L as n grows large. A convergent sequence approaches a finite value, while a divergent sequence does not settle to a single limit. Other properties include monotonicity (increasing or decreasing) and boundedness.

Examples: the arithmetic progression 2, 5, 8, 11,...; the geometric progression 3, 9, 27, 81,...; the Fibonacci sequence defined by F_1 = 1, F_2 = 1, F_n = F_{n−1} + F_{n−2} for n > 2.

Sequences are central in analysis, number theory, and computer science, modeling time series, iterative processes, or data streams. In Icelandic texts, röðunnar is used to discuss properties such as monotonicity, convergence, and related concepts for sequences and the series built from them.