Serien

Serien is the plural form of the German term Serie and is used in mathematics to denote a series: the sum of the terms of a sequence. A series is written as the infinite sum sum_{n=1}^\infty a_n, with partial sums S_N = sum_{n=1}^N a_n. If S_N converges to a finite limit L as N grows, the series is convergent; otherwise it is divergent. Important notions include absolute convergence (sum of |a_n| converges) and conditional convergence. Common examples are geometric series, where a_n = r^n, and power series, where a_n are coefficients of x^n. Various tests, such as the ratio test, root test, comparison test, and the alternating series test, help determine convergence. Infinite series are foundational in analysis and appear in representations of functions, series expansions, and in number theory.

Serien is also the plural of Serie in German and is used to refer to TV series or serialized narratives. In German-language media, fernsehserie denotes a television series; Serien im weiteren Sinn can include web series, radio serials, or serialized magazines. A typical TV series is released in seasons containing multiple episodes and may follow episodic or serial structures, with character development and ongoing plots. The rise of streaming platforms has influenced production, distribution, and audience engagement, with broader access and on-demand viewing shaping how series are produced and consumed. The term also covers literary series and other serialized works published in installments.

Etymology and scope: from Latin series, via French série, referring to order and succession. The concept of Serien captures both mathematical sequences of terms and culturally serialized narratives.