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mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a geometric
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... , where ''a'' is the
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of each term and ''r'' is the common ratio between adjacent terms. Geometric series are among the simplest examples of
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
and can serve as a basic introduction to
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
and
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Geometric series had an important role in the early development of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, are used throughout mathematics, and have important applications in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
,
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

biology
,
economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interact ...

economics
,
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
,
queueing theory Queueing theory is the mathematical study of waiting lines, or wikt:queue, queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research b ...
, and
finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

finance
. The distinction between a progression and a series is that a progression is a sequence, whereas a series is a sum.


Coefficient ''a''

The geometric series ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... is written in expanded form.Riddle, Douglas F. ''Calculus and Analytic Geometry, Second Edition'' Belmont, California, Wadsworth Publishing, p. 566, 1970. Every coefficient in the geometric series is the same. In contrast, the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
written as ''a''0 + ''a''1''r'' + ''a''2''r''2 + ''a''3''r''3 + ... in expanded form has coefficients ''a''i that can vary from term to term. In other words, the geometric series is a
special case In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
of the power series. The first term of a geometric series in expanded form is the coefficient ''a'' of that geometric series. In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as :\sum^_ ''ar''k and a closed form of the geometric series written as :''a'' / (1 - ''r'') within the range , ''r'', < 1. The derivation of the closed form from the expanded form is shown in this article's section. The derivation requires that all the coefficients of the series be the same (coefficient a) in order to take advantage of
self-similarity __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...
and to reduce the infinite number of additions and power operations in the expanded form to the single subtraction and single division in the closed form. However even without that derivation, the result can be confirmed with
long division In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

long division
: ''a'' divided by (1 - ''r'') results in ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... , which is the expanded form of the geometric series. Typically a geometric series is thought of as a sum of numbers ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... but can also be thought of as a sum of functions ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... that converges to the function ''a'' / (1 - ''r'') within the range , r, < 1. The adjacent image shows the contribution each of the first nine terms (i.e., functions) make to the function ''a'' / (1 - ''r'') within the range , ''r'', < 1 when ''a'' = 1. Changing even one of the coefficients to something other than coefficient ''a'' would (in addition to changing the geometric series to a power series) change the resulting sum of functions to some function other than ''a'' / (1 - ''r'') within the range , ''r'', < 1. As an aside, a particularly useful change to the coefficients is defined by the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, which describes how to change the coefficients so that the sum of functions converges to any user selected, sufficiently smooth function within a range.


Common ratio ''r''

The geometric series ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... is an infinite series defined by just two
parameters A parameter (), generally, is any characteristic that can help in defining or classifying a particular system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...

parameters
: coefficient ''a'' and common ratio ''r''. Common ratio ''r'' is the ratio of any term with the previous term in the series. Or equivalently, common ratio ''r'' is the term multiplier used to calculate the next term in the series. The following table shows several geometric series: The convergence of the geometric series depends on the value of the common ratio ''r'': :* If , ''r'', < 1, the terms of the series approach zero in the limit (becoming smaller and smaller in
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...

magnitude
), and the series converges to the sum ''a'' / (1 - ''r''). :* If , ''r'', = 1, the series does not converge. When ''r'' = 1, all of the terms of the series are the same and the series is infinite. When ''r'' = −1, the terms take two values alternately (for example, 2, −2, 2, −2, 2,... ). The sum of the terms
oscillates Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. The term ''vibration'' is precisely used to describe ...
between two values (for example, 2, 0, 2, 0, 2,... ). This is a different type of divergence. See for example
Grandi's seriesIn mathematics, the infinite series , also written : \sum_^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent ...
: 1 − 1 + 1 − 1 + ···. :*If , ''r'', > 1, the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series does not converge to a sum. (The series diverges.) The rate of convergence also depends on the value of the common ratio ''r''. Specifically, the rate of convergence gets slower as ''r'' approaches 1 or −1. For example, the geometric series with ''a'' = 1 is 1 + ''r'' + ''r''2 + ''r''3 + ... and converges to 1 / (1 - ''r'') when , ''r'', < 1. However, the number of terms needed to converge approaches infinity as ''r'' approaches 1 because ''a'' / (1 - ''r'') approaches infinity and each term of the series is less than or equal to one. In contrast, as ''r'' approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of ''r'' that is even or odd. That flipping behavior near ''r'' = −1 is illustrated in the adjacent image showing the first 11 terms of the geometric series with ''a'' = 1 and , ''r'', < 1. The common ratio ''r'' and the coefficient ''a'' also define the
geometric progression In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

geometric progression
, which is a list of the terms of the geometric series but without the additions. Therefore the geometric series ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... has the geometric progression (also called the geometric sequence) ''a'', ''ar'', ''ar''2, ''ar''3, ... The geometric progression - as simple as it is - models a surprising number of natural
phenomena A phenomenon (; plural phenomena) is an observable fact or event. The term came into its modern philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge ...
, :* from some of the largest observations such as the
expansion of the universe The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby ''the scale of space itself changes''. The universe does n ...
where the common ratio ''r'' is defined by
Hubble's constant Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from the Earth at speeds proportional to their distance. In other words, the farther they are the faster they are moving ...
, :* to some of the smallest observations such as the decay of radioactive carbon-14 atoms where the common ratio ''r'' is defined by the half-life of carbon-14. As an aside, the common ratio ''r'' can be a
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
such as , ''r'', ei''θ'' where , ''r'', is the
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
's magnitude (or length), ''θ'' is the vector's angle (or orientation) in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and i2 = -1. With a common ratio , ''r'', ei''θ'', the expanded form of the geometric series is ''a'' + ''a'', ''r'', ei''θ'' + ''a'', ''r'', 2ei2''θ'' + ''a'', ''r'', 3ei3''θ'' + ... Modeling the angle ''θ'' as linearly increasing over time at the rate of some
angular frequency In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
''ω''0 (in other words, making the substitution ''θ'' = ''ω''0''t''), the expanded form of the geometric series becomes ''a'' + ''a'', ''r'', ei''ω''0''t'' + ''a'', ''r'', 2ei2''ω''0''t'' + ''a'', ''r'', 3ei3''ω''0''t'' + ... , where the first term is a vector of length ''a'' not rotating at all, and all the other terms are vectors of different lengths rotating at
harmonics A harmonic is any member of the harmonic series Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig .... The term is employed in various disciplines, including music ...

harmonics
of the fundamental angular frequency ''ω''0. The constraint , ''r'', <1 is enough to coordinate this infinite number of vectors of different lengths all rotating at different speeds into tracing a circle, as shown in the adjacent video. Similar to how the
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
describes how to change the coefficients so the series converges to a user selected sufficiently smooth function within a range, the
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
describes how to change the coefficients (which can also be complex numbers in order to specify the initial angles of vectors) so the series converges to a user selected
periodic function A periodic function is a Function (mathematics), function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used th ...

periodic function
.


Sum


Closed-form formula

For r\neq 1, the sum of the first ''n''+1 terms of a geometric series, up to and including the ''r'' n term, is :a + ar + a r^2 + a r^3 + \cdots + a r^n = \sum_^ ar^k= a \left(\frac\right), where is the common ratio. One can derive that closed-form formula for the partial sum, ''s'', by subtracting out the many
self-similar __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...
terms as follows: : \begin s = a\ +\ &ar\ +\ ar^2\ +\ ar^3\ +\ \cdots\ +\ ar^, \\ rs =\ &ar\ +\ ar^2\ +\ ar^3\ +\ \cdots\ +\ ar^\ +\ ar^, \\ s - rs =\ &a\ -\ ar^, \\ s(1-r) =\ &a (1-r^), \\ s =\ &a \left(\frac\right) \quad \text r \neq 1 \text. \end As approaches infinity, the absolute value of must be less than one for the series to converge. The sum then becomes :a+ar+ar^2+ar^3+ar^4+\cdots = \sum_^\infty ar^k = \frac, \text , r, <1. When , this can be simplified to :1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots \;=\; \frac. The formula also holds for complex , with the corresponding restriction, the modulus of is strictly less than one. As an aside, the question of whether an infinite series converges is fundamentally a question about the distance between two values: given enough terms, does the value of the partial sum get arbitrarily close to the value it is approaching? In the above derivation of the closed form of the geometric series, the interpretation of the distance between two values is the distance between their locations on the
number line In elementary mathematics 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) include ...

number line
. That is the most common interpretation of distance between two values. However the
p-adic group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analys ...
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
, which has become a critical notion in modern
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, offers a definition of distance such that the geometric series 1 + 2 + 4 + 8 + ... with ''a'' = 1 and ''r'' = 2 actually does converge to ''a'' / (1 - ''r'') = 1 / (1 - 2) = -1 even though ''r'' is outside the typical convergence range , ''r'', < 1.


Proof of convergence

We can prove that the geometric series converges using the sum formula for a
geometric progression In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

geometric progression
: :\begin 1 + r + r^2 + r^3 + \cdots \ &= \lim_ \left(1 + r + r^2 + \cdots + r^n\right) \\ &= \lim_ \frac. \end The second equality is true because if , r, < 1, then r^ \to 0 as n \to \infty and : \begin (1 + r + r^2 + \cdots + r^n)(1 - r) &= ((1-r) + (r - r^2) + (r^2 - r^3) + ... + (r^n - r^))\\ &= (1 + (-r + r) + ( -r^2 + r^2) + ... + (-r^n + r^n) - r^)\\ &= 1-r^. \end Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent
telescoping series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Consider the function, : g(K) = \frac. Note that : 1 = g(0) - g(1), \quad r = g(1) - g(2), \quad r^2 = g(2) - g(3), \ldots Thus, : S = 1 + r + r^2 + r^3 + \cdots = (g(0) - g(1)) + (g(1) - g(2)) + (g(2) - g(3)) + \cdots . If : , r, <1 then : g(K)\longrightarrow 0 \text K \to \infty. So ''S'' converges to : g(0) = \frac.


Rate of convergence

As shown in the above proofs, the closed form of the geometric series partial sum up to and including the ''n''-th power of ''r'' is ''a''(1 - ''r''''n''+1) / (1 - ''r'') for any value of ''r'', and the closed form of the geometric series is the full sum ''a'' / (1 - ''r'') within the range , ''r'', < 1. If the common ratio is within the range 0 < ''r'' < 1, then the partial sum ''a''(1 - ''r''''n''+1) / (1 - ''r'') increases with each added term and eventually gets within some small error, ''E'', ratio of the full sum ''a'' / (1 - ''r''). Solving for ''n'' at that error threshold, : \begin a \left(\frac\right) &\geq a \left(\frac\right), \\ 1 - r^ &\geq 1 - E, \\ r^ &\leq E, \\ \ln(r^) &\leq \ln(E), \\ (n+1)\ln(r) &\leq \ln(E), \\ n + 1 &\geq \left\lceil \frac \right\rceil, \end where 0 < ''r'' < 1, the ceiling operation \lceil \rceil constrains ''n'' to integers, and dividing both sides by the natural log of ''r'' flips the inequality because it is negative. The result ''n''+1 is the number of partial sum terms needed to get within ''aE'' / (1 - ''r'') of the full sum ''a'' / (1 - ''r''). For example to get within 1% of the full sum ''a'' / (1 - ''r'') at ''r''=0.1, only 2 (= ln(''E'') / ln(''r'') = ln(0.01) / ln(0.1)) terms of the partial sum are needed. However at ''r''=0.9, 44 (= ln(0.01) / ln(0.9)) terms of the partial sum are needed to get within 1% of the full sum ''a'' / (1 - ''r''). If the common ratio is within the range -1 < ''r'' < 0, then the geometric series is an alternating series but can be converted into the form of a non-alternating geometric series by combining pairs of terms and then analyzing the rate of convergence using the same approach as shown for the common ratio range 0 < ''r'' < 1. Specifically, the partial sum :s = ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ''ar''4 + ''ar''5 + ... + ''ar''''n''-1 + ''ar''''n'' within the range -1 < ''r'' < 0 is equivalent to :s = ''a'' - ''ap'' + ''ap''2 - ''ap''3 + ''ap''4 - ''ap''5 + ... + ''ap''''n''-1 - ''ap''''n'' with an ''n'' that is odd, with the substitution of ''p'' = -''r'', and within the range 0 < ''p'' < 1, :s = (''a'' - ''ap'') + (''ap''2 - ''ap''3) + (''ap''4 - ''ap''5) + ... + (''ap''''n''-1 - ''ap''n) with adjacent and differently signed terms paired together, :s = ''a''(1 - ''p'') + ''a''(1 - ''p'')''p''2 + ''a''(1 - ''p'')''p''4 + ... + ''a''(1 - ''p'')''p''2(''n''-1)/2 with ''a''(1 - ''p'') factored out of each term, :s = ''a''(1 - ''p'') + ''a''(1 - ''p'')''p''2 + ''a''(1 - ''p'')''p''4 + ... + ''a''(1 - ''p'')''p''2''m'' with the substitution ''m'' = (''n'' - 1) / 2 which is an integer given the constraint that ''n'' is odd, which is now in the form of the first ''m'' terms of a geometric series with coefficient ''a''(1 - ''p'') and with common ratio ''p''2. Therefore the closed form of the partial sum is ''a''(1 - ''p'')(1 - ''p''2(''m''+1)) / (1 - ''p''2) which increases with each added term and eventually gets within some small error, ''E'', ratio of the full sum ''a''(1 - ''p'') / (1 - ''p''2). As before, solving for ''m'' at that error threshold, : \begin a(1-p) \left(\frac\right) &\geq a(1-p) \left(\frac\right), \\ 1 - p^ &\geq 1 - E, \\ p^ &\leq E, \\ \ln(p^) &\leq \ln(E), \\ (2m+2)\ln(p) &\leq \ln(E), \\ m + 1 &\geq \left\lceil \frac \right\rceil, \\ m + 1 &\geq \left\lceil \frac \right\rceil, \end where 0 < ''p'' < 1 or equivalently -1 < ''r'' < 0, and the ''m''+1 result is the number of partial sum pairs of terms needed to get within ''a''(1 - ''p'')''E'' / (1 - ''p''2) of the full sum ''a''(1 - ''p'') / (1 - ''p''2). For example to get within 1% of the full sum ''a''(1 - ''p'') / (1 - ''p''2) at ''p''=0.1 or equivalently ''r''=-0.1, only 1 (= ln(''E'') / (2 ln(''p'')) = ln(0.01) / (2 ln(0.1)) pair of terms of the partial sum are needed. However at ''p''=0.9 or equivalently ''r''=-0.9, 22 (= ln(0.01) / (2 ln(0.9))) pairs of terms of the partial sum are needed to get within 1% of the full sum ''a''(1 - ''p'') / (1 - ''p''2). Comparing the rate of convergence for positive and negative values of ''r'', ''n'' + 1 (the number of terms required to reach the error threshold for some positive ''r'') is always twice as large as ''m'' + 1 (the number of term pairs required to reach the error threshold for the negative of that ''r'') but the ''m'' + 1 refers to term pairs instead of single terms. Therefore, the rate of convergence is
symmetric Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

symmetric
about ''r'' = 0, which can be a surprise given the asymmetry of ''a'' / (1 - ''r''). One perspective that helps explain this rate of convergence symmetry is that on the ''r'' > 0 side each added term of the partial sum makes a finite contribution to the infinite sum at ''r'' = 1 while on the ''r'' < 0 side each added term makes a finite contribution to the infinite slope at ''r'' = -1. As an aside, this type of rate of convergence analysis is particularly useful when calculating the number of Taylor series terms needed to adequately approximate some user-selected sufficiently-smooth function or when calculating the number of Fourier series terms needed to adequately approximate some user-selected periodic function.


Historic insights


Zeno of Elea (c.495 – c.430 BC)

2,500 years ago, Greek mathematicians had a problem with walking from one place to another. Physically, they were able to walk as well as we do today, perhaps better. Logically, however, they thought that an infinitely long list of numbers greater than zero summed to infinity. Therefore, it was a paradox when
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Pre-Socratic philosophy is ancient Greek philosophy Ancient Greek philosophy arose in the 6th century BC, at a time when the inhabitants of ancient Greece were st ...

Zeno of Elea
pointed out that in order to walk from one place to another, you first have to walk half the distance, and then you have to walk half the remaining distance, and then you have to walk half of that remaining distance, and you continue halving the remaining distances an infinite number of times because no matter how small the remaining distance is you still have to walk the first half of it. Thus, Zeno of Elea transformed a short distance into an infinitely long list of halved remaining distances, all of which are greater than zero. And that was the problem: how can a distance be short when measured directly and also infinite when summed over its infinite list of halved remainders? The paradox revealed something was wrong with the assumption that an infinitely long list of numbers greater than zero summed to infinity.


Euclid of Alexandria (c.300 BC)

''Euclid's Elements of Geometry'' Book IX, Proposition 35, proof (of the proposition in adjacent diagram's caption): The terseness of Euclid's propositions and proofs may have been a necessity. As is, the ''Elements of Geometry'' is over 500 pages of propositions and proofs. Making copies of this popular textbook was labor intensive given that the
printing press A printing press is a mechanical device for applying pressure to an ink Ink is a gel, sol, or solution Image:SaltInWaterSolutionLiquid.jpg, Making a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in wate ...
was not invented until 1440. And the book's popularity lasted a long time: as stated in the cited introduction to an English translation, ''Elements of Geometry'' "has the distinction of being the world's oldest continuously used mathematical textbook." So being very terse was being very practical. The proof of Proposition 35 in Book IX could have been even more compact if Euclid could have somehow avoided explicitly equating lengths of specific line segments from different terms in the series. For example, the contemporary notation for geometric series (i.e., ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... + ''ar''n) does not label specific portions of terms that are equal to each other. Also in the cited introduction the editor comments,
Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems (e.g., Theorem 48 in Book 1).
To help translate the proposition and proof into a form that uses current notation, a couple modifications are in the diagram. First, the four horizontal line lengths representing the values of the first four terms of a geometric series are now labeled a, ar, ar2, ar3 in the diagram's left margin. Second, new labels A' and D' are now on the first and third lines so that all the diagram's line segment names consistently specify the segment's starting point and ending point. Here is a phrase by phrase interpretation of the proposition: Similarly, here is a sentence by sentence interpretation of the proof:


Archimedes of Syracuse (c.287 – c.212 BC)

Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
used the sum of a geometric series to compute the area enclosed by a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

parabola
and a straight line. His method was to dissect the area into an infinite number of triangles. Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle. Archimedes 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. Assuming that the blue triangle has area 1, the total area is an infinite sum: :1 \,+\, 2\left(\frac\right) \,+\, 4\left(\frac\right)^2 \,+\, 8\left(\frac\right)^3 \,+\, \cdots. The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives :1 \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots. This is a geometric series with common ratio and the fractional part is equal to :\sum_^\infty 4^ = 1 + 4^ + 4^ + 4^ + \cdots = . The sum is :\frac\;=\;\frac\;=\;\frac. This computation uses the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
, an early version of . Using
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, the same area could be found by a
definite integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
.


Nicole Oresme (c.1323 – 1382)

Among his insights into infinite series, in addition to his elegantly simple proof of the divergence of the harmonic series,
Nicole Oresme Nicole Oresme (; c. 1320–1325 – 11 July 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a French philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of ge ...
proved that the series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. His diagram for his geometric proof, similar to the adjacent diagram, shows a two dimensional geometric series. The first dimension is horizontal, in the bottom row showing the geometric series ''S'' = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient ''a'' = 1/2 and common ratio ''r'' = 1/2 that converges to ''S'' = ''a'' / (1-''r'') = (1/2) / (1-1/2) = 1. The second dimension is vertical, where the bottom row is a new coefficient ''a''''T'' equal to ''S'' and each subsequent row above it is scaled by the same common ratio ''r'' = 1/2, making another geometric series ''T'' = 1 + 1/2 + 1/4 + 1/8 + ... , which is the geometric series with coefficient ''a''''T'' = ''S'' = 1 and common ratio ''r'' = 1/2 that converges to ''T'' = ''a''''T'' / (1-''r'') = ''S'' / (1-''r'') = ''a'' / (1-''r'') / (1-''r'') = (1/2) / (1-1/2) / (1-1/2) = 2. Although difficult to visualize beyond three dimensions, Oresme's insight generalizes to any dimension ''d''. Using the sum of the ''d''−1 dimension of the geometric series as the coefficient ''a'' in the ''d'' dimension of the geometric series results in a ''d''-dimensional geometric series converging to ''S''''d'' / ''a'' = 1 / (1-''r'')''d'' within the range , ''r'', <1.
Pascal's triangle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
and long division reveals the coefficients of these multi-dimensional geometric series, where the closed form is valid only within the range , ''r'', <1. :\begin \text \\ \text \quad \text \\ \text \quad \text \quad \text \\ \text \quad\text \quad \text \quad\text \\ \text \quad\text \quad \text \quad \text \quad \text \end :\begin &d \quad S^d / a\ \text \quad &&S^d / a\ \text \\ &1 \quad 1 / (1-r) \quad &&1 + r + r^2 + r^3 + r^4 + \cdots \\ &2 \quad 1 / (1-r)^2 \quad &&1 + 2r + 3r^2 + 4r^3 + 5r^4 + \cdots \\ &3 \quad 1 / (1-r)^3 \quad &&1 + 3r + 6r^2 + 10r^3 + 15r^4 + \cdots \\ &4 \quad 1 / (1-r)^4 \quad &&1 + 4r + 10r^2 + 20r^3 + 35r^4 + \cdots \\ \end Note that as an alternative to long division, it is also possible to calculate the coefficients of the ''d''-dimensional geometric series by integrating the coefficients of dimension ''d''−1. This mapping from division by 1-''r'' in the power series sum domain to integration in the power series coefficient domain is a discrete form of the mapping performed by the
Laplace transform In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable t (often time in physics, time) to a function of a complex analysis, complex variable s (co ...
. MIT Professor Arthur Mattuck shows how to derive the Laplace transform from the power series in this lecture video, where the power series is a mapping between discrete coefficients and a sum and the Laplace transform is a mapping between continuous weights and an integral. The closed forms of ''Sd''/''a'' are related to but not equal to the derivatives of S = f(''r'') = 1 / (1-''r''). As shown in the following table, the relationship is ''S''''k''+1 = f(''k'')(''r'') / ''k''!, where f(''k'')(''r'') denotes the ''k''th derivative of f(''r'') = 1 / (1-''r'') and the closed form is valid only within the range , ''r'', < 1. :\begin &k \quad f^(r) / a\ \text \quad &&f^(r) / a\ \text \\ &0 \quad 1 / (1-r) \quad &&\sum^_r^ \\ &1 \quad 1! / (1-r)^2 \quad &&\sum^_j r^ \\ &2 \quad 2! / (1-r)^3 \quad &&\sum^_j(j-1) r^ \\ &3 \quad 3! / (1-r)^4 \quad &&\sum^_j(j-1)(j-2) r^ \\ &4 \quad 4! / (1-r)^5 \quad &&\sum^_j(j-1)(j-2)(j-3) r^ \\ \end


Applications


Repeating decimals

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example: :0.7777\ldots \;=\; \frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots. The formula for the sum of a geometric series can be used to convert the decimal to a fraction, :0.7777\ldots \;=\; \frac \;=\; \frac \;=\; \frac \;=\; \frac. The formula works not only for a single repeating figure, but also for a repeating group of figures. For example: :0.123412341234\ldots \;=\; \frac \;=\; \frac \;=\; \frac \;=\; \frac. Note that every series of repeating consecutive decimals can be conveniently simplified with the following: :0.09090909\ldots \;=\; \frac \;=\; \frac. :0.143814381438\ldots \;=\; \frac. :0.9999\ldots \;=\; \frac \;=\; 1. That is, a repeating decimal with repeat length is equal to the quotient of the repeating part (as an integer) and .


Economics

In
economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interact ...

economics
, geometric series are used to represent the
present value In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behavi ...
of an
annuity An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account A savings acco ...
(a sum of money to be paid in regular intervals). For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in
perpetuity A perpetuity is an annuity An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings ...
. Receiving $100 a year from now is worth less than an immediate $100, because one cannot
invest Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance Finance is the study of financial institution ...

invest
the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + I ), where I is the yearly interest rate. Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + I)2 (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is :\sum_^\infty \frac, which is the infinite series: :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots. This is a geometric series with common ratio 1 / (1 + I ). The sum is the first term divided by (one minus the common ratio): :\frac \;=\; \frac. For example, if the yearly interest rate is 10% (I = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000. This sort of calculation is used to compute the APR of a loan (such as a
mortgage loan A mortgage loan or simply mortgage () is a loan In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money ...
). It can also be used to estimate the present value of expected , or the terminal value of a
financial asset A financial asset is a non-physical asset In financial accounting Financial accounting is the field of accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information abou ...
assuming a stable growth rate.


Fractal geometry

In the study of
fractal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

fractal
s, geometric series often arise as the
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which t ...

perimeter
,
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
, or
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
of a
self-similar __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...
figure. For example, the area inside the
Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...

Koch snowflake
can be described as the union of infinitely many
equilateral triangle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

equilateral triangle
s (see figure). Each 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. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is :1 \,+\, 3\left(\frac\right) \,+\, 12\left(\frac\right)^2 \,+\, 48\left(\frac\right)^3 \,+\, \cdots. The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio ''r'' = 4/9. The first term of the geometric series is ''a'' = 3(1/9) = 1/3, so the sum is :1\,+\,\frac\;=\;1\,+\,\frac\;=\;\frac. Thus the Koch snowflake has 8/5 of the area of the base triangle.


Geometric power series

The formula for a geometric series :\frac=1+x+x^2+x^3+x^4+\cdots can be interpreted as a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
in the
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
sense, converging where , x, <1. From this, one can extrapolate to obtain other power series. For example, :\begin \tan^(x)&=\int\frac\\ &=\int\frac\\ &=\int\left(1 + \left(-x^2\right) + \left(-x^2\right)^2 + \left(-x^2\right)^3+\cdots\right)dx\\ &=\int\left(1-x^2+x^4-x^6+\cdots\right)dx\\ &=x-\frac+\frac-\frac+\cdots\\ &=\sum^_ \frac x^. \end


See also

* * * * * * * * * *


Specific geometric series

* : 1 − 1 + 1 − 1 + ⋯ * * * * * * A geometric series is a unit series (the series sum converges to one) if and only if , ''r'', < 1 and ''a'' + ''r'' = 1 (equivalent to the more familiar form S = ''a'' / (1 - ''r'') = 1 when , ''r'', < 1). Therefore, an
alternating series Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alternati ...
is also a unit series when -1 < ''r'' < 0 and ''a'' + ''r'' = 1 (for example, coefficient ''a'' = 1.7 and common ratio ''r'' = -0.7). * The terms of a geometric series are also the terms of a generalized
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omi ...

Fibonacci sequence
(Fn = Fn-1 + Fn-2 but without requiring F0 = 0 and F1 = 1) when a geometric series common ratio ''r'' satisfies the constraint 1 + ''r'' = ''r''2, which according to the
quadratic formula In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

quadratic formula
is when the common ratio ''r'' equals the
golden ratio In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

golden ratio
(i.e., common ratio ''r'' = (1 ± √5)/2). * The only geometric series that is a unit series and also has terms of a generalized
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omi ...

Fibonacci sequence
has the
golden ratio In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

golden ratio
as its coefficient ''a'' and the conjugate
golden ratio In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

golden ratio
as its common ratio ''r'' (i.e., ''a'' = (1 + √5)/2 and ''r'' = (1 - √5)/2). It is a unit series because ''a'' + ''r'' = 1 and , ''r'', < 1, it is a generalized
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omi ...

Fibonacci sequence
because 1 + ''r'' = ''r''2, and it is an
alternating series Alternating may refer to: Mathematics * Alternating algebra, an algebra in which odd-grade elements square to zero * Alternating form, a function formula in algebra * Alternating group, the group of even permutations of a finite set * Alternati ...
because ''r'' < 0.


Notes


References

* Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972. * * Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 278–279, 1985. * Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987. * Courant, R. and Robbins, H. "The Geometric Progression." §1.2.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 13–14, 1996. * James Stewart (2002). ''Calculus'', 5th ed., Brooks Cole. * Larson, Hostetler, and Edwards (2005). ''Calculus with Analytic Geometry'', 8th ed., Houghton Mifflin Company. * * Pappas, T. "Perimeter, Area & the Infinite Series." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 134–135, 1989. * * Roger B. Nelsen (1997). ''Proofs without Words: Exercises in Visual Thinking'', The Mathematical Association of America.


History and philosophy

* C. H. Edwards Jr. (1994). ''The Historical Development of the Calculus'', 3rd ed., Springer. . * *
Eli Maor Eli Maor (born 1937), an Israel-born historian of mathematics, is the author of several books about the history of mathematics. Eli Maor received his PhD at the Technion – Israel Institute of Technology The Technion – Israel Institute of Tec ...
(1991). ''To Infinity and Beyond: A Cultural History of the Infinite'', Princeton University Press. * Morr Lazerowitz (2000). ''The Structure of Metaphysics (International Library of Philosophy)'', Routledge.


Economics

* Carl P. Simon and Lawrence Blume (1994). ''Mathematics for Economists'', W. W. Norton & Company. * Mike Rosser (2003). ''Basic Mathematics for Economists'', 2nd ed., Routledge.


Biology

* Edward Batschelet (1992). ''Introduction to Mathematics for Life Scientists'', 3rd ed., Springer. * Richard F. Burton (1998). ''Biology by Numbers: An Encouragement to Quantitative Thinking'', Cambridge University Press.


Computer science

* John Rast Hubbard (2000). ''Schaum's Outline of Theory and Problems of Data Structures With Java'', McGraw-Hill.


External links

* * * * *
"Geometric Series"
by Michael Schreiber,
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hoste ...
, 2007. {{Authority control Ratios Articles containing proofs