Numerosarjat

Numerosarjat, or number sequences, are ordered lists of numbers in which each term is indexed by a positive integer. They are usually denoted (a_n) and can be defined by explicit formulas or by recursive rules. The study includes properties such as convergence, monotonicity, and boundedness, and their relation to sums of terms, known as series.

Arithmetic sequences have a constant difference d between successive terms, with a_n = a_1 + (n−1)d. Example: 5, 8, 11, 14, ... with d = 3. Geometric sequences have a constant ratio r between successive terms, with a_n = a_1 r^(n−1). Example: 2, 6, 18, 54, ... with r = 3.

Convergence concerns the behavior of a sequence as n grows large. A sequence (a_n) converges to a limit L if a_n approaches L as n → ∞; otherwise it diverges. Sequences can be bounded and monotone, and if a sequence is increasing and bounded above, it converges to a finite limit.

Series are the sums of the terms of a sequence. The nth partial sum is S_n = a_1 + ... + a_n, and an infinite series may converge to a finite value. Convergence tests, such as the ratio test or comparison tests, assess whether a series converges, and distinctions are made between absolute and conditional convergence.

Numerosarjat appear in many areas of mathematics and its applications. They model patterns, support numerical approximations, and help in understanding functions and algorithms through structured, rule-based sequences.