Lukusarjan

Lukusarjan is the Finnish term used for a numerical sequence: an ordered list of numbers (a1, a2, a3, ...) in which each term is indexed by a natural number and defined by a rule, explicit formula, or recurrence relation. A sequence can be given directly by a_n = f(n) or by a relationship among its terms, such as a_n = a_{n-1} + d with a starting value.

Common examples include arithmetic sequences, where the difference between consecutive terms is constant (a_n = a1 + (n−1)d), and geometric sequences, where each term is a constant multiple of the previous (a_n = a1 r^{n−1}). The Fibonacci sequence is a well-known nontrivial example, defined by a_n = a_{n-1} + a_{n-2} with initial values a1 = a2 = 1.

Convergence to a limit is a central idea in the study of lukusarjat. A sequence (a_n) converges to L if the terms get arbitrarily close to L as n grows, formalized by the requirement that for every ε > 0 there exists N with |a_n − L| < ε for all n ≥ N. Sequences may be monotone and bounded, properties that influence convergence, and they can have divergent behavior as well.

When a sequence is used to form a sum, it generates a tersine, or series, through partial sums S_N = sum_{k=1}^N a_k. Series convergence is a related but distinct concept, analyzed with tests such as the ratio test or comparison test.

Lukusarjat appear in mathematics, numerical analysis, statistics (time series), and computer science, serving as foundational objects for modeling, computation, and theoretical analysis. See also sequence, series, limit.