Editing Geometric series (section)

== Definition and examples ==
The geometric series is an [[infinite series]] derived from a special type of sequence called a [[geometric progression]]. This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term <math> a </math>, and the next one being the initial term multiplied by a constant number known as the common ratio <math> r </math>. By multiplying each term with a common ratio continuously, the geometric series can be defined mathematically as{{r|vpr}}
<math display="block"> a + ar + ar^2 + ar^3 + \cdots = \sum_{k=0}^\infty ar^k. </math>
The sum of a finite initial segment of an infinite geometric series is called a '''finite geometric series''', expressed as{{r|young}}
<math display="block"> a + ar + ar^2 + ar^3 + \cdots + ar^n = \sum_{k=0}^n ar^k. </math>

When <math>r > 1</math> it is often called a growth rate or rate of expansion. When <math>0 < r < 1</math> it is often called a decay rate or shrink rate, where the idea that it is a "rate" comes from interpreting <math>k</math> as a sort of discrete time variable. When an application area has specialized vocabulary for specific types of growth, expansion, shrinkage, and decay, that vocabulary will also often be used to name <math>r</math> parameters of geometric series. In [[economics]], for instance, rates of increase and decrease of [[price level]]s are called [[inflation]] rates and [[deflation]] rates, while rates of increase in [[Value (economics)|values]] of [[investment]]s include [[Rate of return|rates of return]] and [[interest rate]]s.{{r|cz}}

[[File:GeometricSquares.svg|thumb|upright=1|The geometric series <math display="inline"> \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \cdots </math> shown as areas of purple squares. Each of the purple squares has {{sfrac|1|4}} of the area of the next larger square <math display="inline"> \frac{1}{2} \times 
\frac{1}{2} = \frac{1}{4} </math>, <math display="inline"> \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} </math>, and so forth. Thus, the sum of the purple squares' area is one-third of the area of the large square.]]
When summing infinitely many terms, the geometric series can either be convergent or [[Divergent geometric series|divergent]]. Convergence means there is a value after summing infinitely many terms, whereas divergence means no value after summing. The convergence of a geometric series can be described depending on the value of a common ratio, see {{slink||Convergence of the series and its proof}}. [[Grandi's series]] is an example of a divergent series that can be expressed as <math> 1 - 1 + 1 - 1 + \cdots </math>, where the initial term is <math> 1 </math> and the common ratio is <math> -1 </math>; this is because it has three different values.

[[Decimal]] numbers that have [[Repeating decimal|repeated patterns that continue forever]] can be interpreted as geometric series and thereby converted to expressions of the [[Rational number|ratio of two integers]].{{sfnp|Apostol|1967|p=393}} For example, the repeated decimal fraction <math>0.7777\ldots</math> can be written as the geometric series <math display="block"> 0.7777\ldots = \frac{7}{10} + \frac{7}{10} \left(\frac{1}{10}\right) + \frac{7}{10} \left(\frac{1}{10^2}\right) + \frac{7}{10} \left(\frac{1}{10^3}\right) + \cdots,</math> where the initial term is <math>a = \tfrac{7}{10}</math> and the common ratio is <math>r = \tfrac{1}{10}</math>.