Aufsummung

Aufsummung is the operation of combining a collection of quantities by addition. In mathematics it covers both finite and infinite collections. For a finite list a1, a2, ..., an, the Aufsummung is the sum S = a1 + a2 + ... + an, typically written concisely as S = sum_{i=1}^n a_i.

For an infinite sequence (a_i), one defines partial sums S_N = sum_{i=1}^N a_i and the infinite sum as S = lim_{N→∞} S_N, provided this limit exists. If the limit exists, the series is convergent; if not, the series diverges. The value of a convergent infinite sum can depend on convergence properties, and some series may converge conditionally or absolutely.

Key properties include linearity and index handling. Summation is linear: sum_{i}(α_i x_i) can be distributed as a sum of scaled sums. For finite sums, the order of terms does not matter due to commutativity and associativity of addition. For infinite sums, rearranging or regrouping terms can affect the sum unless the series converges absolutely.

Notations and extensions include multiple sums, such as Σ_i Σ_j a_{ij}, which generalize summation over multi-index sets. Summation is central in various mathematical domains, including arithmetic and geometric series, telescoping sums, and Fourier or power series.

Applications span probability (expectations as sums), combinatorics, numerical analysis, and physics, where discrete accumulations of quantities are modeled via Aufsummung.