Editing Geometric series (section)

{{Short description|Sum of an (infinite) geometric progression}}
{{Calculus |Series}}
In [[mathematics]], a '''geometric series''' is a [[series (mathematics)|series]] summing the terms of an infinite [[geometric sequence]], in which the ratio of consecutive terms is constant. For example, [[1/2 + 1/4 + 1/8 + 1/16 + ⋯|the series <math>\tfrac12 + \tfrac14 + \tfrac18 + \cdots</math>]] is a geometric series with common ratio {{tmath|\tfrac12}}, which converges to the sum of {{tmath|1}}. Each term in a geometric series is the [[geometric mean]] of the term before it and the term after it, in the same way that each term of an [[arithmetic series]] is the [[arithmetic mean]] of its neighbors.

While [[Ancient Greek philosophy|Greek philosopher]] [[Zeno's paradoxes]] about time and motion (5th century BCE) have been interpreted as involving geometric series, such series were formally studied and applied a century or two later by [[Greek mathematics|Greek mathematicians]], for example used by [[Archimedes]] to [[Quadrature of the Parabola|calculate the area inside a parabola]] (3rd century BCE). Today, geometric series are used in [[mathematical finance]], calculating areas of fractals, and various computer science topics.

Though geometric series most commonly involve [[Real number|real]] or [[complex number]]s, there are also important results and applications for [[Matrix (mathematics)|matrix-valued]] geometric series, function-valued geometric series, [[P-adic number|{{nowrap|1=<math>p</math>-}}adic number]] geometric series, and most generally geometric series of elements of abstract algebraic [[Field (mathematics)|field]]s, [[Ring (mathematics)|ring]]s, and [[semiring]]s.