{"history":[],"article":{"id":"c0bafd0a-1440-4898-b58e-7ad281d69c76","title":"Geometric series","slug":"geometric-series","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **geometric [series](https://bliptext.com/articles/series-mathematics)** is a [series](https://bliptext.com/articles/series-mathematics) summing the terms of an infinite [geometric sequence](https://bliptext.com/articles/geometric-sequence), in which the ratio of consecutive terms is constant.\nFor example, [the series ](https://bliptext.com/articles/1-2-1-4-1-8-1-16)is a geometric series with common ratio \\tfrac12, which converges to the sum of 1.\nEach term in a geometric series is the [geometric mean](https://bliptext.com/articles/geometric-mean) of the term before it and the term after it, in the same way that each term of an [arithmetic series](https://bliptext.com/articles/arithmetic-series) is the [arithmetic mean](https://bliptext.com/articles/arithmetic-mean) of its neighbors.\nWhile [Greek philosopher](https://bliptext.com/articles/ancient-greek-philosophy) [Zeno's paradoxes](https://bliptext.com/articles/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 mathematicians](https://bliptext.com/articles/greek-mathematics), for example used by [Archimedes](https://bliptext.com/articles/archimedes) to [calculate the area inside a parabola](https://bliptext.com/articles/quadrature-of-the-parabola) (3rd century BCE).\nToday, geometric series are used in [mathematical finance](https://bliptext.com/articles/mathematical-finance), calculating areas of fractals, and various computer science topics.\nThough geometric series most commonly involve [real](https://bliptext.com/articles/real-number) or [complex numbers](https://bliptext.com/articles/complex-number), there are also important results and applications for [matrix-valued](https://bliptext.com/articles/matrix-mathematics) geometric series, function-valued geometric series, [p-adic number](https://bliptext.com/articles/p-adic-number) geometric series, and most generally geometric series of elements of abstract algebraic [fields](https://bliptext.com/articles/field-mathematics), [rings](https://bliptext.com/articles/ring-mathematics), and semi[rings](https://bliptext.com/articles/ring-mathematics).\n\n# Definition and examples\nThe geometric series is an [infinite series](https://bliptext.com/articles/infinite-series) derived from a special type of sequence called a [geometric progression](https://bliptext.com/articles/geometric-progression).\nThis means that it is the sum of infinitely many terms of geometric progression: starting from the initial term a, and the next one being the initial term multiplied by a constant number known as the common ratio r. By multiplying each term with a common ratio continuously, the geometric series can be defined mathematically as:\nTruncating the geometric series into several terms is called finite geometric series, that is:\nWhen r > 1 it is often called a growth rate or rate of expansion.\nWhen 0 < r < 1 it is often called a decay rate or shrink rate, where the idea that it is a \"rate\" comes from interpreting k as a sort of discrete time variable.\nWhen an application area has specialized vocabulary for specific types of growth, expansion, shrinkage, and decay, that vocabulary will also often be used to name r parameters of geometric series.\nIn [economics](https://bliptext.com/articles/economics), for instance, rates of increase and decrease of [price levels](https://bliptext.com/articles/price-level) are called [inflation](https://bliptext.com/articles/inflation) rates and [deflation](https://bliptext.com/articles/deflation) rates, while rates of increase in [values](https://bliptext.com/articles/value-economics) of [investments](https://bliptext.com/articles/investment) include [rates of return](https://bliptext.com/articles/rate-of-return) and [interest rates](https://bliptext.com/articles/interest-rate).\nWhen summing infinitely many terms, the geometric series can either be convergence or [divergence](https://bliptext.com/articles/divergent-geometric-series).\nConvergence means there is a value after summing infinitely many terms, whereas divergence means no value after summing.\nThe convergence of a geometric series can be described depending on the value of a common ratio, see.\n[Grandi's series](https://bliptext.com/articles/grandi-s-series) is an example of diverge series that can be expressed as, where the initial term is 1 and the common ratio is -1; this because it has three different values.\n[Decimal](https://bliptext.com/articles/decimal) numbers that have [repeated patterns that continue forever](https://bliptext.com/articles/repeating-decimal) can be interpreted as geometric series and thereby converted to expressions of the [ratio of two integers](https://bliptext.com/articles/rational-number).\nFor example, the repeated decimal fraction can be written as the geometric series where the initial term is a = 7 / 10 and the common ratio is r = 1 / 10.\n# [The geometric series shown as areas of purple squares. Each of the purple squares has 1/4 of the area of the next larger square,, and so forth. Thus, the sum of the purple squares' area is one-third of the area of the large square. | upload.wikimedia.org/wikipedia/commons/e/e6/GeometricSquares.svg]\n\n# Convergence of the series and its proof\nThe convergence of the infinite sequence of partial sums of the infinite geometric series depends on the [magnitude](https://bliptext.com/articles/magnitude-mathematics) of the common ratio r alone:\nThe [rate of convergence](https://bliptext.com/articles/rate-of-convergence) shows how the sequence quickly approaches its limit.\nIn the case of the geometric series—the relevant sequence is S_n and its limit is S—the rate and order are found via\nwhere q represents the order of convergence.\nUsing and choosing the order of convergence q = 1 gives:\nWhen the series converges, the rate of convergence gets slower as |r| approaches 1.\nThe pattern of convergence also depends on the [sign](https://bliptext.com/articles/sign-mathematics) or [complex argument](https://bliptext.com/articles/argument-complex-analysis) of the common ratio.\nIf r > 0 and |r| < 1 then terms all share the same sign and the partial sums of the terms approach their eventual limit [monotonically](https://bliptext.com/articles/monotonic-sequence).\nIf r < 0 and |r| < 1, adjacent terms in the geometric series alternate between positive and negative, and the partial sums S_n of the terms oscillate above and below their eventual limit S. For complex r and |r| < 1, the S_n converge in a spiraling pattern.\nThe convergence is proved as follows.\nThe partial sum of the first n + 1 terms of a geometric series, up to and including the r^{n} term,\nis given by the closed form\nwhere r is the common ratio.\nThe case r = 1 is merely a simple addition, a case of an [arithmetic series](https://bliptext.com/articles/arithmetic-series).\nThe formula for the partial sums S_n with r \\neq 1 can be derived as follows:\nfor r \\neq 1.\nAs r approaches 1, polynomial division or [L'Hospital's rule](https://bliptext.com/articles/l-h-pital-s-rule) recovers the case.\nAs n approaches infinity, the absolute value of\nr\nmust be less than one for this sequence of partial sums to converge to a limit.\nWhen it does, the series [converges absolutely](https://bliptext.com/articles/absolute-convergence).\nThe infinite series then becomes\nfor |r| < 1.\nThis convergence result is widely applied to prove the convergence of other series as well, whenever those series's terms can be bounded from above by a suitable geometric series; that proof strategy is the basis for the [ratio test](https://bliptext.com/articles/ratio-test) and [root test](https://bliptext.com/articles/root-test) for the convergence of infinite series.\n# [< 1 and, the r^n term vanishes, leaving . This figure uses a slightly different convention for S_n than the main text, shifted by one term. | upload.wikimedia.org/wikipedia/commons/f/fa/Geometric///progression///sum///visual///proof.svg]\n\n# Connection to the power series\nLike the geometric series, a [power series](https://bliptext.com/articles/power-series) has one parameter for a common variable raised to successive powers corresponding to the geometric series's r, but it has additional parameters one for each term in the series, for the distinct coefficients of each, rather than just a single additional parameter a for all terms, the common coefficient of r^k in each term of a geometric series.\nThe geometric series can therefore be considered a class of power series in which the sequence of coefficients satisfies a_k = a for all k and x = r.\nThis special class of power series plays an important role in mathematics, for instance for the study of [ordinary generating functions](https://bliptext.com/articles/ordinary-generating-functions) in combinatorics and the [summation](https://bliptext.com/articles/summation-method) of divergent series in analysis.\nMany other power series can be written as transformations and combinations of geometric series, making the geometric series formula a convenient tool for calculating formulas for those power series as well.\nAs a power series, the geometric series has a [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of 1.\nThis could be seen as a consequence of the [Cauchy–Hadamard theorem](https://bliptext.com/articles/cauchy-hadamard-theorem) and the fact that for any a or as a consequence of the [ratio test](https://bliptext.com/articles/ratio-test) for the convergence of infinite series, with implying convergence only for |r| < 1.\nHowever, both the ratio test and the Cauchy–Hadamard theorem are proven using the geometric series formula as a logically prior result, so such reasoning would be subtly circular.\n\n# Background\n2,500 years ago, Greek mathematicians believed that an infinitely long list of positive numbers must sum to infinity.\nTherefore, [Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) created a [paradox](https://bliptext.com/articles/zeno-s-paradoxes), demonstrating as follows: in order to walk from one place to another, one must first walk half the distance there, and then half of the remaining distance, and half of that remaining distance, and so on, covering infinitely many intervals before arriving.\nIn doing so, he partitioned a fixed distance into an infinitely long list of halved remaining distances, each with a length greater than zero.\nZeno's paradox revealed to the Greeks that their assumption about an infinitely long list of positive numbers needing to add up to infinity was incorrect.\nEuclid's [Elements](https://bliptext.com/articles/euclid-s-elements) has the distinction of being the world's oldest continuously used mathematical textbook, and it includes a demonstration of the sum of finite geometric series in Book IX, Proposition 35, illustrated in an adjacent figure.\n[Archimedes](https://bliptext.com/articles/archimedes) in his [The Quadrature of the Parabola](https://bliptext.com/articles/the-quadrature-of-the-[parabola](https://bliptext.com/articles/parabola)) used the sum of a geometric series to compute the area enclosed by a [parabola](https://bliptext.com/articles/parabola) and a straight line.\nArchimedes' theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.\nHis method was to dissect the area into infinite triangles as shown in the adjacent figure.\nHe determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.\nAssuming that the blue triangle has area 1, then, the total area is the sum of the infinite series\nHere the first term represents the area of the blue triangle, the second term is the area of the two green triangles, the third term is the area of the four yellow triangles, and so on.\nSimplifying the fractions gives\na geometric series with common ratio r = 1/4 and its sum is:\nIn addition to his elegantly simple proof of the divergence of the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics), [Nicole Oresme](https://bliptext.com/articles/nicole-oresme) proved that the [arithmetico-geometric series](https://bliptext.com/articles/arithmetico-geometric-series) known as Gabriel's Staircase,\nIn the diagram for his geometric proof, similar to the adjacent diagram, shows a two-dimensional geometric series.\nThe first dimension is horizontal, in the bottom row, representing the geometric series with initial value a=1/2 and common ratio r=1/2\nThe second dimension is vertical, where the bottom row is a new initial term a = S and each subsequent row above it shrinks according to the same common ratio r=1/2, making another geometric series with sum T,\nThis approach generalizes usefully to higher dimensions, and that generalization is described below in.\n# [A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. | upload.wikimedia.org/wikipedia/commons/c/c6/Oresme///diagram///of///two///dimensional///geometric///series.png]\n\n# Applications\nAs mentioned above, the geometric series can be applied in the field of [economics](https://bliptext.com/articles/economics).\nThis leads to the common ratio of a geometric series that may refer to the rates of increase and decrease of [price levels](https://bliptext.com/articles/price-level) are called [inflation](https://bliptext.com/articles/inflation) rates and [deflation](https://bliptext.com/articles/deflation) rates; in contrast, the rates of increase in [values](https://bliptext.com/articles/value-economics) of [investments](https://bliptext.com/articles/investment) include [rates of return](https://bliptext.com/articles/rates-of-return) and [interest rates](https://bliptext.com/articles/interest-rates).\nMore specifically in [mathematical finance](https://bliptext.com/articles/mathematical-finance), geometric series can also be applied in [time value of money](https://bliptext.com/articles/time-value-of-money); that is to represent the [present values](https://bliptext.com/articles/present-value) of [perpetual annuities](https://bliptext.com/articles/perpetuity), sums of money to be paid each year indefinitely into the future.\nThis sort of calculation is used to compute the [annual percentage rate](https://bliptext.com/articles/annual-percentage-rate) of a loan, such as a [mortgage loan](https://bliptext.com/articles/mortgage-loan).\nIt can also be used to estimate the present value of expected [stock dividends](https://bliptext.com/articles/dividend), or the [terminal value](https://bliptext.com/articles/terminal-value-finance) of a [financial asset](https://bliptext.com/articles/financial-asset) assuming a stable growth rate.\nHowever, 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.\nIn addition to finding the area enclosed by a parabola and a line in [Archimedes](https://bliptext.com/articles/archimedes)' [The Quadrature of the Parabola](https://bliptext.com/articles/the-quadrature-of-the-parabola), the geometric series may also be applied in finding the [Koch snowflake](https://bliptext.com/articles/koch-snowflake)'s area described as the union of infinitely many [equilateral triangles](https://bliptext.com/articles/equilateral-triangle) (see figure).\nEach side of the green triangle is exactly 1/3 the size of a side of the large blue triangle and therefore has exactly 1/9 the area.\nSimilarly, each yellow triangle has 1/9 the area of a green triangle, and so forth.\nAll 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.\nExcluding the initial 1, this series has a common ratio, and by taking the blue triangle as a unit of area, the total area of the snowflake is:\nVarious topics in computer science may include the application of geometric series in the following:\n# [The interior of the Koch snowflake is a union of infinitely many triangles | upload.wikimedia.org/wikipedia/commons/e/ed/Koch///Snowflake///Triangles.png]\n\n# Beyond real and complex numbers\nWhile geometric series with real and complex number parameters a and r are most common, geometric series of more general terms such as [functions](https://bliptext.com/articles/function-mathematics), [matrices](https://bliptext.com/articles/matrix-mathematics), and [p-adic numbers](https://bliptext.com/articles/p-adic-number) also find application.\nThe 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 [modern algebra](https://bliptext.com/articles/abstract-algebra), to define geometric series with parameters from any [ring](https://bliptext.com/articles/ring-mathematics) or [field](https://bliptext.com/articles/field-mathematics).\nFurther generalization to geometric series with parameters from [semirings](https://bliptext.com/articles/semiring) is more unusual, but also has applications; for instance, in the study of [fixed-point iteration](https://bliptext.com/articles/fixed-point-iteration) of [transformation functions](https://bliptext.com/articles/transformation-function), as in transformations of [automata](https://bliptext.com/articles/automata-theory) via [rational series](https://bliptext.com/articles/rational-series).\nIn order to analyze the convergence of these general geometric series, then on top of addition and multiplication, one must also have some [metric of distance](https://bliptext.com/articles/metric-space) between partial sums of the series.\nThis can introduce new subtleties into the questions of convergence, such as the distinctions between [uniform convergence](https://bliptext.com/articles/uniform-convergence) and [pointwise convergence](https://bliptext.com/articles/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 + ...](https://bliptext.com/articles/1-2-4-8) with a=1 and r = 2 to in the [2-adic numbers](https://bliptext.com/articles/p-adic-number) using the [2-adic absolute value](https://bliptext.com/articles/p-adic-valuation) as a convergence metric.\nIn that case, the 2-adic absolute value of the common coefficient is, and while this is counterintuitive from the perspective of real number absolute value (where |2| = 2, naturally), it is nonetheless well-justified in the context of [p-adic analysis](https://bliptext.com/articles/p-adic-analysis).\nWhen the multiplication of the parameters is not [commutative](https://bliptext.com/articles/commutative-property), as it often is not for matrices or general [physical operators](https://bliptext.com/articles/operator-physics), particularly in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), then the standard way of writing the geometric series,, multiplying from the right, may need to be distinguished from the alternative , multiplying from the left, and also the symmetric , multiplying half on each side.\nThese 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 [Ito integration](https://bliptext.com/articles/it-calculus) and [Stratonovitch integration](https://bliptext.com/articles/stratonovich-integral) in [stochastic calculus](https://bliptext.com/articles/stochastic-calculus).\n\n## History and philosophy\n\n## Economics\n\n## Biology","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:20:55.736Z","updated_at":"2025-01-25T17:20:55.736Z","search_vector":"'geometr':1 'seri':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **geometric [series](https://bliptext.com/articles/series-mathematics)** is a [series](https://bliptext.com/articles/series-mathematics) summing the terms of an infinite [geometric sequence](https://bliptext.com/articles/geometric-sequence), in which the ratio of consecutive terms is constant.\nFor example, [the series ](https://bliptext.com/articles/1-2-1-4-1-8-1-16)is a geometric series with common ratio \\tfrac12, which converges to the sum of 1.\nEach term in a geometric series is the [geometric mean](https://bliptext.com/articles/geometric-mean) of the term before it and the term after it, in the same way that each term of an [arithmetic series](https://bliptext.com/articles/arithmetic-series) is the [arithmetic mean](https://bliptext.com/articles/arithmetic-mean) of its neighbors.\nWhile [Greek philosopher](https://bliptext.com/articles/ancient-greek-philosophy) [Zeno's paradoxes](https://bliptext.com/articles/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 mathematicians](https://bliptext.com/articles/greek-mathematics), for example used by [Archimedes](https://bliptext.com/articles/archimedes) to [calculate the area inside a parabola](https://bliptext.com/articles/quadrature-of-the-parabola) (3rd century BCE).\nToday, geometric series are used in [mathematical finance](https://bliptext.com/articles/mathematical-finance), calculating areas of fractals, and various computer science topics.\nThough geometric series most commonly involve [real](https://bliptext.com/articles/real-number) or [complex numbers](https://bliptext.com/articles/complex-number), there are also important results and applications for [matrix-valued](https://bliptext.com/articles/matrix-mathematics) geometric series, function-valued geometric series, [p-adic number](https://bliptext.com/articles/p-adic-number) geometric series, and most generally geometric series of elements of abstract algebraic [fields](https://bliptext.com/articles/field-mathematics), [rings](https://bliptext.com/articles/ring-mathematics), and semi[rings](https://bliptext.com/articles/ring-mathematics).\n\n# Definition and examples\nThe geometric series is an [infinite series](https://bliptext.com/articles/infinite-series) derived from a special type of sequence called a [geometric progression](https://bliptext.com/articles/geometric-progression).\nThis means that it is the sum of infinitely many terms of geometric progression: starting from the initial term a, and the next one being the initial term multiplied by a constant number known as the common ratio r. By multiplying each term with a common ratio continuously, the geometric series can be defined mathematically as:\nTruncating the geometric series into several terms is called finite geometric series, that is:\nWhen r > 1 it is often called a growth rate or rate of expansion.\nWhen 0 0 and |r| < 1 then terms all share the same sign and the partial sums of the terms approach their eventual limit [monotonically](https://bliptext.com/articles/monotonic-sequence).\nIf r < 0 and |r| < 1, adjacent terms in the geometric series alternate between positive and negative, and the partial sums S_n of the terms oscillate above and below their eventual limit S. For complex r and |r| < 1, the S_n converge in a spiraling pattern.\nThe convergence is proved as follows.\nThe partial sum of the first n + 1 terms of a geometric series, up to and including the r^{n} term,\nis given by the closed form\nwhere r is the common ratio.\nThe case r = 1 is merely a simple addition, a case of an [arithmetic series](https://bliptext.com/articles/arithmetic-series).\nThe formula for the partial sums S_n with r \\neq 1 can be derived as follows:\nfor r \\neq 1.\nAs r approaches 1, polynomial division or [L'Hospital's rule](https://bliptext.com/articles/l-h-pital-s-rule) recovers the case.\nAs n approaches infinity, the absolute value of\nr\nmust be less than one for this sequence of partial sums to converge to a limit.\nWhen it does, the series [converges absolutely](https://bliptext.com/articles/absolute-convergence).\nThe infinite series then becomes\nfor |r| < 1.\nThis convergence result is widely applied to prove the convergence of other series as well, whenever those series's terms can be bounded from above by a suitable geometric series; that proof strategy is the basis for the [ratio test](https://bliptext.com/articles/ratio-test) and [root test](https://bliptext.com/articles/root-test) for the convergence of infinite series.\n# [< 1 and, the r^n term vanishes, leaving . This figure uses a slightly different convention for S_n than the main text, shifted by one term. | upload.wikimedia.org/wikipedia/commons/f/fa/Geometric///progression///sum///visual///proof.svg]\n\n# Connection to the power series\nLike the geometric series, a [power series](https://bliptext.com/articles/power-series) has one parameter for a common variable raised to successive powers corresponding to the geometric series's r, but it has additional parameters one for each term in the series, for the distinct coefficients of each, rather than just a single additional parameter a for all terms, the common coefficient of r^k in each term of a geometric series.\nThe geometric series can therefore be considered a class of power series in which the sequence of coefficients satisfies a_k = a for all k and x = r.\nThis special class of power series plays an important role in mathematics, for instance for the study of [ordinary generating functions](https://bliptext.com/articles/ordinary-generating-functions) in combinatorics and the [summation](https://bliptext.com/articles/summation-method) of divergent series in analysis.\nMany other power series can be written as transformations and combinations of geometric series, making the geometric series formula a convenient tool for calculating formulas for those power series as well.\nAs a power series, the geometric series has a [radius of convergence](https://bliptext.com/articles/radius-of-convergence) of 1.\nThis could be seen as a consequence of the [Cauchy–Hadamard theorem](https://bliptext.com/articles/cauchy-hadamard-theorem) and the fact that for any a or as a consequence of the [ratio test](https://bliptext.com/articles/ratio-test) for the convergence of infinite series, with implying convergence only for |r| < 1.\nHowever, both the ratio test and the Cauchy–Hadamard theorem are proven using the geometric series formula as a logically prior result, so such reasoning would be subtly circular.\n\n# Background\n2,500 years ago, Greek mathematicians believed that an infinitely long list of positive numbers must sum to infinity.\nTherefore, [Zeno of Elea](https://bliptext.com/articles/zeno-of-elea) created a [paradox](https://bliptext.com/articles/zeno-s-paradoxes), demonstrating as follows: in order to walk from one place to another, one must first walk half the distance there, and then half of the remaining distance, and half of that remaining distance, and so on, covering infinitely many intervals before arriving.\nIn doing so, he partitioned a fixed distance into an infinitely long list of halved remaining distances, each with a length greater than zero.\nZeno's paradox revealed to the Greeks that their assumption about an infinitely long list of positive numbers needing to add up to infinity was incorrect.\nEuclid's [Elements](https://bliptext.com/articles/euclid-s-elements) has the distinction of being the world's oldest continuously used mathematical textbook, and it includes a demonstration of the sum of finite geometric series in Book IX, Proposition 35, illustrated in an adjacent figure.\n[Archimedes](https://bliptext.com/articles/archimedes) in his [The Quadrature of the Parabola](https://bliptext.com/articles/the-quadrature-of-the-[parabola](https://bliptext.com/articles/parabola)) used the sum of a geometric series to compute the area enclosed by a [parabola](https://bliptext.com/articles/parabola) and a straight line.\nArchimedes' theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.\nHis method was to dissect the area into infinite triangles as shown in the adjacent figure.\nHe determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.\nAssuming that the blue triangle has area 1, then, the total area is the sum of the infinite series\nHere the first term represents the area of the blue triangle, the second term is the area of the two green triangles, the third term is the area of the four yellow triangles, and so on.\nSimplifying the fractions gives\na geometric series with common ratio r = 1/4 and its sum is:\nIn addition to his elegantly simple proof of the divergence of the [harmonic series](https://bliptext.com/articles/harmonic-series-mathematics), [Nicole Oresme](https://bliptext.com/articles/nicole-oresme) proved that the [arithmetico-geometric series](https://bliptext.com/articles/arithmetico-geometric-series) known as Gabriel's Staircase,\nIn the diagram for his geometric proof, similar to the adjacent diagram, shows a two-dimensional geometric series.\nThe first dimension is horizontal, in the bottom row, representing the geometric series with initial value a=1/2 and common ratio r=1/2\nThe second dimension is vertical, where the bottom row is a new initial term a = S and each subsequent row above it shrinks according to the same common ratio r=1/2, making another geometric series with sum T,\nThis approach generalizes usefully to higher dimensions, and that generalization is described below in.\n# [A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. | upload.wikimedia.org/wikipedia/commons/c/c6/Oresme///diagram///of///two///dimensional///geometric///series.png]\n\n# Applications\nAs mentioned above, the geometric series can be applied in the field of [economics](https://bliptext.com/articles/economics).\nThis leads to the common ratio of a geometric series that may refer to the rates of increase and decrease of [price levels](https://bliptext.com/articles/price-level) are called [inflation](https://bliptext.com/articles/inflation) rates and [deflation](https://bliptext.com/articles/deflation) rates; in contrast, the rates of increase in [values](https://bliptext.com/articles/value-economics) of [investments](https://bliptext.com/articles/investment) include [rates of return](https://bliptext.com/articles/rates-of-return) and [interest rates](https://bliptext.com/articles/interest-rates).\nMore specifically in [mathematical finance](https://bliptext.com/articles/mathematical-finance), geometric series can also be applied in [time value of money](https://bliptext.com/articles/time-value-of-money); that is to represent the [present values](https://bliptext.com/articles/present-value) of [perpetual annuities](https://bliptext.com/articles/perpetuity), sums of money to be paid each year indefinitely into the future.\nThis sort of calculation is used to compute the [annual percentage rate](https://bliptext.com/articles/annual-percentage-rate) of a loan, such as a [mortgage loan](https://bliptext.com/articles/mortgage-loan).\nIt can also be used to estimate the present value of expected [stock dividends](https://bliptext.com/articles/dividend), or the [terminal value](https://bliptext.com/articles/terminal-value-finance) of a [financial asset](https://bliptext.com/articles/financial-asset) assuming a stable growth rate.\nHowever, 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.\nIn addition to finding the area enclosed by a parabola and a line in [Archimedes](https://bliptext.com/articles/archimedes)' [The Quadrature of the Parabola](https://bliptext.com/articles/the-quadrature-of-the-parabola), the geometric series may also be applied in finding the [Koch snowflake](https://bliptext.com/articles/koch-snowflake)'s area described as the union of infinitely many [equilateral triangles](https://bliptext.com/articles/equilateral-triangle) (see figure).\nEach side of the green triangle is exactly 1/3 the size of a side of the large blue triangle and therefore has exactly 1/9 the area.\nSimilarly, each yellow triangle has 1/9 the area of a green triangle, and so forth.\nAll 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.\nExcluding the initial 1, this series has a common ratio, and by taking the blue triangle as a unit of area, the total area of the snowflake is:\nVarious topics in computer science may include the application of geometric series in the following:\n# [The interior of the Koch snowflake is a union of infinitely many triangles | upload.wikimedia.org/wikipedia/commons/e/ed/Koch///Snowflake///Triangles.png]\n\n# Beyond real and complex numbers\nWhile geometric series with real and complex number parameters a and r are most common, geometric series of more general terms such as [functions](https://bliptext.com/articles/function-mathematics), [matrices](https://bliptext.com/articles/matrix-mathematics), and [p-adic numbers](https://bliptext.com/articles/p-adic-number) also find application.\nThe 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 [modern algebra](https://bliptext.com/articles/abstract-algebra), to define geometric series with parameters from any [ring](https://bliptext.com/articles/ring-mathematics) or [field](https://bliptext.com/articles/field-mathematics).\nFurther generalization to geometric series with parameters from [semirings](https://bliptext.com/articles/semiring) is more unusual, but also has applications; for instance, in the study of [fixed-point iteration](https://bliptext.com/articles/fixed-point-iteration) of [transformation functions](https://bliptext.com/articles/transformation-function), as in transformations of [automata](https://bliptext.com/articles/automata-theory) via [rational series](https://bliptext.com/articles/rational-series).\nIn order to analyze the convergence of these general geometric series, then on top of addition and multiplication, one must also have some [metric of distance](https://bliptext.com/articles/metric-space) between partial sums of the series.\nThis can introduce new subtleties into the questions of convergence, such as the distinctions between [uniform convergence](https://bliptext.com/articles/uniform-convergence) and [pointwise convergence](https://bliptext.com/articles/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 + ...](https://bliptext.com/articles/1-2-4-8) with a=1 and r = 2 to in the [2-adic numbers](https://bliptext.com/articles/p-adic-number) using the [2-adic absolute value](https://bliptext.com/articles/p-adic-valuation) as a convergence metric.\nIn that case, the 2-adic absolute value of the common coefficient is, and while this is counterintuitive from the perspective of real number absolute value (where |2| = 2, naturally), it is nonetheless well-justified in the context of [p-adic analysis](https://bliptext.com/articles/p-adic-analysis).\nWhen the multiplication of the parameters is not [commutative](https://bliptext.com/articles/commutative-property), as it often is not for matrices or general [physical operators](https://bliptext.com/articles/operator-physics), particularly in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), then the standard way of writing the geometric series,, multiplying from the right, may need to be distinguished from the alternative , multiplying from the left, and also the symmetric , multiplying half on each side.\nThese 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 [Ito integration](https://bliptext.com/articles/it-calculus) and [Stratonovitch integration](https://bliptext.com/articles/stratonovich-integral) in [stochastic calculus](https://bliptext.com/articles/stochastic-calculus).\n\n## History and philosophy\n\n## Economics\n\n## Biology","revisionCount":0},"hasMore":false,"nextCursor":null}