Editing Geometric series (section)

== Beyond real and complex numbers ==
While geometric series with real and complex number parameters <math>a</math> and <math>r</math> are most common, geometric series of more general terms such as [[Function (mathematics)|functions]], [[Matrix (mathematics)|matrices]], and [[P-adic number|{{nowrap|1=<math>p</math>-}}adic number]]s also find application.{{r|robert}} The mathematical operations used to express a geometric series given its parameters are simply addition and repeated multiplication, and so it is natural, in the context of [[Abstract algebra|modern algebra]], to define geometric series with parameters from any [[Ring (mathematics)|ring]] or [[Field (mathematics)|field]].{{r|df}} Further generalization to geometric series with parameters from [[semiring]]s is more unusual, but also has applications; for instance, in the study of [[fixed-point iteration]] of [[Transformation (function)|transformation functions]], as in transformations of [[Automata theory|automata]] via [[rational series]].{{r|kulch}}

In order to analyze the convergence of these general geometric series, then on top of addition and multiplication, one must also have some [[Metric space|metric of distance]] between partial sums of the series. This can introduce new subtleties into the questions of convergence, such as the distinctions between [[uniform convergence]] and [[pointwise convergence]] in series of functions, and can lead to strong contrasts with intuitions from the real numbers, such as in the convergence of the series [[1 + 2 + 4 + 8 + ⋯|<math>1+2+4+8+\cdots</math>]] with <math>a=1</math> and <math>r = 2</math> to <math display="block">\frac{a }{ 1-r } = -1</math> in the [[P-adic number|2-adic numbers]] using the [[P-adic valuation|2-adic absolute value]] as a convergence metric. In that case, the 2-adic absolute value of the common coefficient is <math>|r|_2 = |2|_2 = \tfrac12</math>, and while this is counterintuitive from the perspective of real number absolute value (where <math>|2| = 2,</math> naturally), it is nonetheless well-justified in the context of [[p-adic analysis]].{{r|robert}}

When the multiplication of the parameters is not [[Commutative property|commutative]], as it often is not for matrices or general [[Operator (physics)|physical operators]], particularly in [[quantum mechanics]], then the standard way of writing the geometric series,

<math display="block">a + ar + ar^2 + ar^3 + \cdots,</math>

multiplying from the right, may need to be distinguished from the alternative

<math display="block">a + ra + r^2a + r^3a + \cdots,</math>

multiplying from the left, and also the symmetric

<math display="block">a + r^\frac12 ar^\frac12 + rar + r^\frac32 ar^\frac32 + \cdots,</math>

multiplying half on each side. These choices may correspond to important alternatives with different strengths and weaknesses in applications, as in the case of ordering the mutual interferences of drift and diffusion differently at infinitesimal temporal scales in [[Itô calculus|Ito integration]] and [[Stratonovich integral|Stratonovitch integration]] in [[stochastic calculus]].