reihe

Reihe is a German noun with several related meanings. In everyday language it denotes a line, row, or sequence of items arranged in order, such as a Reihe von Häusern or a sequence of events. It also appears in publishing or retail to mean a book series or a product line.

In mathematics, Reihe has a specialized meaning: a series, the sum of the terms of a sequence. A sequence, or Folge, is an ordered list of numbers indexed by natural numbers; the associated Reihe is the infinite sum ∑ a_n. A Reihe may converge to a finite value or diverge. Partial sums S_N = ∑_{n=1}^N a_n are studied to determine convergence. Common examples include the geometric series ∑_{n=0}^\infty ar^n, which converges if |r| < 1, and the harmonic series ∑_{n=1}^\infty 1/n, which diverges. In German mathematical writing, Folge (sequence) and Reihe (series) are distinct concepts, though the distinction may be blurred for non-specialists.

Other uses: outside mathematics, Reihe frequently means a row of objects (for example, eine Reihe von Bäumen). It also appears in media and publishing to denote a series of works, such as a Buchreihe (book series) or a TV series. Translations depend on context, with Reihe commonly rendered as row, line, sequence, or series in English.