series

In mathematics, a series is the sum of the terms of a sequence. If a sequence {a_n} is given, the corresponding series is the sequence of partial sums S_N = sum_{n=1}^N a_n; an infinite series is the limit S = lim_{N→∞} S_N, if it exists. If the limit exists, the series converges to S; otherwise it diverges.

Convergence and divergence are central concerns. Convergence tests assess whether S_N tends to a finite limit. Common tests include the ratio test, the root test, the integral test, and comparison tests; the alternating series test applies to alternating terms. A series is absolutely convergent if sum |a_n| converges; otherwise it may converge conditionally. Absolute convergence implies convergence regardless of term rearrangement; conditional convergence can be affected by rearrangement (Riemann series theorem).

Important families include the geometric series sum a r^n, which converges to a/(1−r) when |r|<1. The harmonic series sum 1/n diverges; p-series sum 1/n^p converge iff p>1. Telescoping series have partial sums that cancel to a simple form.

Series are used to represent functions. Power series ∑ a_n x^n have a radius of convergence and can represent analytic functions near a center. Special series include Fourier series, which express periodic functions as sums of sines and cosines, and Taylor or Maclaurin series, which expand functions around a point.

In other contexts, “series” can refer to a sequence of related works released sequentially, such as a television series or a book series.