Sumrekke

Sumrekke is a concept in mathematics used to describe the sum of a sequence of numbers. Given a sequence a1, a2, a3, ..., the sumrekke, or series, is the expression ∑_{n=1}^∞ a_n. The sequence of partial sums S_N = ∑_{n=1}^N a_n records the accumulated total after N terms. The central question is whether the limit lim_{N→∞} S_N exists and is finite. If it does, the sumrekke converges to that limit; otherwise it diverges.

Finite sums are often analyzed through their partial sums, while infinite sums are studied for convergence. Special forms include arithmetic and geometric series. For an arithmetic sequence with first term a_1 and difference d, the partial sum is S_n = n/2 (2a_1 + (n - 1)d) or S_n = n/2 (a_1 + a_n). For a geometric sequence with first term a_1 and ratio r, S_n = a_1 (1 - r^n)/(1 - r) if r ≠ 1, and the infinite geometric series converges to a_1/(1 - r) when |r| < 1.

Convergence tests determine whether a sumrekke converges. Common criteria include the comparison test, ratio test, root test, and the alternating-series test. Examples: the harmonic series ∑ 1/n diverges; the alternating harmonic series ∑ (-1)^{n+1}/n converges conditionally; a telescoping series such as ∑ (1/n - 1/(n+1)) equals 1.

Sumrekkes are central in analysis and appear in applications ranging from solving equations to representing functions via series. They also underpin many numerical methods and theoretical results in mathematics.