Reeksen

Reeksen is a Dutch term used in mathematics to refer to two related concepts: sequences and series. A sequence (reeks) is an ordered list of objects, typically numbers, indexed by the natural numbers: a1, a2, a3, ... The value to which the terms approach, if any, is called the limit of the sequence. If a_n tends to L as n grows, the sequence is convergent; otherwise it diverges. Sequences can be monotone (nondecreasing or nonincreasing) and can be bounded or unbounded.

Series (telreeks) are sums of the terms of a sequence: a1 + a2 + a3 + ... The partial sums s_N = a1 + ... + a_N form another sequence, and the series is convergent when s_N tends to a finite limit S. The value S is then called the sum of the series. Common tests determine convergence, such as the ratio test, root test, comparison test, and, for alternating series, the alternating series test. If the terms decrease in absolute value to zero and the series converges, it may be absolutely convergent or conditionally convergent.

Special kinds include arithmetic reeksen, where successive terms differ by a constant, and geometric reeksen, where successive terms are multiplied by a constant ratio. For a geometric series with ratio r, the sum is S = a1/(1 - r) when |r| < 1.

Reeksen are foundational in real analysis, enabling definitions of power series, Fourier series, and numerical methods. They also underpin many convergence concepts used to study functions, sequences, and series in calculus and beyond.