szereg

Szereg is a Polish term with several related meanings in mathematics and everyday language. In mathematics, szereg denotes the sum of the terms of a sequence. If (a_n) is a sequence, the finite szereg S_N = a_1 + a_2 + … + a_N, and an infinite szereg is defined as the limit of S_N as N → ∞, when that limit exists; the limit is called the sum of the szereg. A main distinction is that a ciąg is the ordered list of terms, while a szereg concerns their accumulated sum.

Common types include:

- szereg arytmetyczny, the sum of an arithmetic progression; for an initial term a_1 and difference d, the finite sum is S_N = N/2 (2a_1 + (N-1)d);

- szereg geometryczny, the sum of a geometric progression; if the common ratio r satisfies |r| < 1, the infinite sum is S = a_1/(1 - r);

- szereg harmoniczny, the series 1/n, which diverges;

- szereg potęgowy (power series) sum c_n x^n, studied for radius of convergence.

Szereg czasowy, or time series, refers to observations indexed in time and is central in statistics and econometrics for analyzing trends and seasonality.

In everyday Polish usage, szereg also means a row, line or sequence of objects or actions, e.g., szereg domów, szereg miejsc, or a military formation in a line.