Editing Geometric series (section)

== Applications ==
As mentioned above, the geometric series can be applied in the field of [[economics]]. This leads to the common ratio of a geometric series that may refer to the rates of increase and decrease of [[price level]]s are called [[inflation]] rates and [[deflation]] rates; in contrast, the rates of increase in [[Value (economics)|value]]s of [[investment]]s include [[rates of return]] and [[interest rates]]. More specifically in [[mathematical finance]], geometric series can also be applied in [[time value of money]]; that is to represent the [[present value]]s of [[Perpetuity|perpetual annuities]], sums of money to be paid each year indefinitely into the future. This sort of calculation is used to compute the [[annual percentage rate]] of a loan, such as a [[mortgage loan]]. It can also be used to estimate the present value of expected [[Dividend|stock dividends]], or the [[Terminal value (finance)|terminal value]] of a [[financial asset]] assuming a stable growth rate. However, the assumption that interest rates are constant is generally incorrect and payments are unlikely to continue forever since the issuer of the perpetual annuity may lose its ability or end its commitment to make continued payments, so estimates like these are only heuristic guidelines for [[decision making]] rather than scientific predictions of actual current values.{{r|cz}}

[[File:Koch Snowflake Triangles.png|thumb|upright|The interior of the [[Koch snowflake]] is a union of infinitely many triangles]]
In addition to finding the area enclosed by a parabola and a line in [[Archimedes]]' ''[[The Quadrature of the Parabola]]'',{{r|sd}} the geometric series may also be applied in finding the [[Koch snowflake]]'s area described as the union of infinitely many [[equilateral triangle]]s (see figure). Each side of the green triangle is exactly {{sfrac|1|3}} the size of a side of the large blue triangle and therefore has exactly {{sfrac|1|9}} the area.  Similarly, each yellow triangle has {{sfrac|1|9}} the area of a green triangle, and so forth. All of these triangles can be represented in terms of geometric series: the blue triangle's area is the first term, the three green triangles' area is the second term, the twelve yellow triangles' area is the third term, and so forth. Excluding the initial 1, this series has a common ratio <math display="inline"> r = \frac{4}{9} </math>, and by taking the blue triangle as a unit of area, the total area of the snowflake is:{{r|kl}}
<math display="block"> 1 + 3\left(\frac{1}{9}\right) + 12\left(\frac{1}{9}\right)^2 + 48\left(\frac{1}{9}\right)^3 + \cdots = \frac{1}{1 - \frac{4}{9}} = \frac{8}{5}.</math>

Various topics in computer science may include the application of geometric series in the following:{{cn|date=November 2024}}
* Algorithm analysis: analyzing the [[time complexity]] of [[Recursion|recursive]] algorithms (like divide-and-conquer) and in [[amortized analysis]] for operations with varying costs, such as dynamic [[Array (data structure)|array]] resizing.
* Data structures: analyzing the [[Computational complexity|space and time complexities]] of operations in data structures like balanced [[binary search tree]]s and [[Heap (data structure)|heaps]].
* Computer graphics: crucial in [[Rendering (computer graphics)|rendering]] algorithms for [[anti-aliasing]], for [[mipmap]]ping, and for generating [[fractal]]s, where the scale of detail varies geometrically.
* Networking and communication: modelling retransmission delays in [[exponential backoff]] algorithms and are used in [[data compression]] and [[Error correction code|error-correcting codes]] for efficient communication.
* Probabilistic and randomized algorithms: analyzing [[random walk]]s, [[Markov chain]]s, and [[geometric distribution]]s, which are essential in probabilistic and [[randomized algorithm]]s.