sommazione

Sommazione is the mathematical operation of adding together a sequence of numbers, expressions, or functions to obtain a total. In Italian, sommazione denotes the process of summing, while the term summation or sigma notation is used in English texts. The concept is central to the study of sequences and series.

Notation and types. The standard notation uses the sigma symbol Σ. A finite sum is written as sum_{k=m}^n a_k, where the index k runs from m to n. An infinite sum, or series, is written as sum_{k=0}^\infty a_k. Finite sums are common in algebra and combinatorics, while infinite sums underpin many topics in analysis, probability, and mathematical physics.

Examples. A finite sum: sum_{k=1}^4 k = 10. An infinite series: sum_{k=1}^\infty (1/2)^k = 1. Not all infinite sums converge; convergence depends on the terms a_k and is studied through various tests and criteria.

Properties. Key properties include linearity: sum (a_k + b_k) = sum a_k + sum b_k, and sum c a_k = c sum a_k. Index shifts are often used: sum_{k=m}^n a_k = sum_{k=m+1}^{n+1} a_{k-1}. Interchanging the order of summation is valid under conditions such as absolute convergence for double sums.

Applications and context. Sommazione is foundational in calculus, analysis, and applied fields, enabling the evaluation of polynomials, generating functions, Fourier and power series, and probabilistic expectations. It connects to related concepts like sequences, series, and various summation formulas.