szorozat

Szorozat is a mathematical concept referring to the operation of multiplication, yielding a product from a collection of elements. In Hungarian mathematical usage, it denotes the act or result of multiplying factors, and it appears in various contexts from elementary arithmetic to higher algebra.

Notation and finite products: For a finite family of numbers a1, a2, ..., an, the szorozat is the product a1 × a2 × ... × an. It is commonly written in compact form as ∏_{i=1}^n a_i. For an indexed family (a_i)_{i∈I}, the product is ∏_{i∈I} a_i. In real or complex numbers, multiplication is associative and commutative, so the order of factors does not affect the result; in other algebraic structures, multiplication may be non-commutative.

Infinite products: An infinite szorozat is denoted ∏_{n=1}^∞ a_n and is defined as the limit of the partial products P_N = ∏_{n=1}^N a_n, if this limit exists. Convergence criteria vary with the context; for positive terms, a common test uses the convergence of the sum ∑ ln(a_n). If any factor is zero, the infinite product is zero; otherwise, an infinite product can converge to a nonzero limit or diverge.

Contexts and examples: The concept appears throughout mathematics. In number theory and combinatorics, products arise in factorials and multiplicative functions. In analysis, products relate to special functions and infinite products representations. In linear algebra, the product of matrices is a central operation and is generally non-commutative. In abstract algebra, multiplication defines the binary operation of a monoid or semiring.

Examples include n! = 1 × 2 × ... × n and the product of a geometric progression a, ar, ar^2, ..., ar^{n-1} equals a^n r^{n(n-1)/2}.