szeregi

Szeregi is the Polish term for the mathematical concept known in English as series. It describes the sum of the terms of a sequence. A finite series consists of a finite number of terms and has a definite finite value. An infinite series sums an infinite sequence of terms and is studied through its sequence of partial sums S_N = sum_{k=1}^N a_k.

An infinite series may converge or diverge. Convergence means S_N approaches a finite limit as N grows. Divergence means no finite limit exists. Several tests help determine this: the geometric series test (sum a r^k converges to a/(1-r) if |r|<1); the p-series test (sum 1/n^p converges if p>1 and diverges if p≤1); the harmonic series diverges; the alternating series test; the ratio test; the root test; the integral test. Absolute convergence is when sum |a_n| converges; otherwise convergence may be conditional, in which rearrangements can affect the sum.

Common representations include power series sum c_n x^n, generating functions, and Fourier series in analysis. In many applications, closed forms exist for particular series (for example, the geometric series). Some series do not have elementary sums, but can be studied through limits, approximations, or special functions such as the Riemann zeta function.