szeregu

Szeregu is the genitive form of the Polish noun szeregu, which in mathematics corresponds to the English term series. In everyday Polish usage, szeregu can also refer more generally to an arrangement or line of objects, but in mathematical contexts it denotes the sum of the terms of a sequence. The article typically centers on the mathematical meaning: a series is the sum of the terms of a sequence, expressed as ∑ a_n, and is studied through its sequence of partial sums S_N = ∑_{n=1}^N a_n.

A central concept is convergence. An infinite series converges if the sequence of its partial sums (S_N) has a finite limit as N grows without bound; the limit, when it exists, is called the sum of the series. If no such limit exists, the series diverges. This framework supports a variety of convergence tests and classifications. Classic examples include the geometric series ∑ a r^{n-1}, which converges when |r| < 1 and has sum a/(1−r), and the harmonic series ∑ 1/n, which diverges. Power series, of the form ∑ c_n x^n, are another important class, characterized by a radius of convergence in the variable x.

In Polish mathematical literature, phrases such as granica szeregu (limit of a series) and szereg potęgowy (power series) are common, reflecting how szeregu is embedded in analytic and algebraic contexts. Beyond analysis, the word szeregu can appear in non-mathematical senses to denote an ordered row or sequence, but the technical meaning relevant to szereg u’s inflection remains the mathematical series and its convergence properties.