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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a geometric series 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, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
of each term and $r$ is the common ratio between adjacent terms. The geometric series had an important role in the early development of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...
, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
, the complex
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ' ...
, and the
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
. The name geometric series indicates each term is the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as th ...
of its two neighboring terms, similar to how the name arithmetic series indicates each term is the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The col ...
of its two neighboring terms. The
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of 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 ...
terms (without any of the additions) is called a ''
geometric sequence In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
'' or ''geometric progression''.

# Formulation

## 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 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
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, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case i ...
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 :$\frac \text , r, <1.$ The derivation of the closed form from the expanded form is shown in this article's Sum section. However even without that derivation, the result can be confirmed with long division: ''a'' divided by (1 - ''r'') results in ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... , which is the expanded form of the geometric series. It is often a convenience in notation to set the series equal to the sum ''s'' and work with the geometric series :''s'' = ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ''ar''4 + ... in its normalized form :''s'' / ''a'' = 1 + ''r'' + ''r''2 + ''r''3 + ''r''4 + ... or in its normalized vector form :''s'' / ''a'' = 1 1 1 1 ...1 ''r'' ''r''2 ''r''3 ''r''4 ...]T or in its normalized partial series form :''s''n / ''a'' = 1 + ''r'' + ''r''2 + ''r''3 + ''r''4 + ... + ''r''n, where n is the power (or degree) of the last term included in the partial sum ''s''n. Changing even one of the coefficients to something other than coefficient ''a'' would 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, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
, 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 (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
: 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), 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 or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
between two values (for example, 2, 0, 2, 0, 2,... ). This is a different type of divergence. See for example Grandi's series: 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, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
, 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 ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfr ...
, :* 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 not e ...
where the common ratio ''r'' is defined by Hubble's constant, :* 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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
such as , ''r'', ei''θ'' where , ''r'', is the
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathemat ...
's magnitude (or length), ''θ'' is the vector's angle (or orientation) in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
''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 a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
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, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
describes how to change the coefficients so the series converges to a user selected sufficiently smooth function within a range, the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ' ...
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 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 throughout science to des ...
.

# Sum

The sum of the first ''n'' terms of a geometric series, up to and including the ''r'' n-1 term, is given by the closed-form formula: $\begin s_n &= ar^0 + ar^1 + \cdots + ar^\\ &= \sum_^ ar^k = \sum_^ ar^\\ &= \begin a\left(\frac\right), \text r \neq 1\\ an, \text r = 1 \end \end$ where is the common ratio. One can derive that closed-form formula for the partial sum, ''s''n, by subtracting out the many
self-similar __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
terms as follows: $\begin s_n &= ar^0 + ar^1 + \cdots + ar^,\\ rs_n &= ar^1 + ar^2 + \cdots + ar^,\\ s_n - rs_n &= ar^0 - ar^,\\ s_n\left(1-r\right) &= a\left(1-r^\right),\\ s_n &= a\left(\frac\right), \text r \neq 1. \end$ As approaches infinity, the absolute value of must be less than one for the series to converge. The sum then becomes $\begin s &= a+ar+ar^2+ar^3+ar^4+\cdots\\ &= \sum_^\infty ar^ = \sum_^\infty ar^\\ &= \frac, \text , r, <1. \end$ The formula also holds for complex , with the corresponding restriction that 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 finite 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, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. That is the most common interpretation of the distance between two values. However, the
p-adic In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
, which has become a critical notion in modern
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, 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, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
: :$\begin 1 + r + r^2 + r^3 + \cdots \ &= \lim_ \left\left(1 + r + r^2 + \cdots + r^n\right\right) \\ &= \lim_ \frac. \end$ The second equality is true because if $, r, < 1,$ then $r^ \to 0$ as $n \to \infty$ and :$\begin \left(1 + r + r^2 + \cdots + r^n\right)\left(1 - r\right) &= \left(\left(1-r\right) + \left(r - r^2\right) + \left(r^2 - r^3\right) + ... + \left(r^n - r^\right)\right)\\ &= \left(1 + \left(-r + r\right) + \left( -r^2 + r^2\right) + ... + \left(-r^n + r^n\right) - r^\right)\\ &= 1-r^. \end$ Alternatively, a geometric interpretation of the convergence is shown in the adjacent diagram. The area of the white triangle is the series
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In alge ...
= ''s'' - ''s''n = ''ar''n+1 / (1 - ''r''). Each additional term in the partial series reduces the area of that white triangle remainder by the area of the trapezoid representing the added term. The trapezoid areas (i.e., the values of the terms) get progressively thinner and shorter and closer to the origin. In the limit, as the number of trapezoids approaches infinity, the white triangle remainder vanishes as it is filled by trapezoids and therefore ''s''n converges to ''s'', provided , ''r'', <1. In contrast, if , ''r'', >1, the trapezoid areas representing the terms of the series instead get progressively wider and taller and farther from the origin, not converging to the origin and not converging as a series.

## Rate of convergence

After knowing that a series converges, there are some applications in which it is also important to know how quickly the series converges. For the geometric series, one convenient measure of the convergence rate is how much the previous series remainder decreases due to the last term of the partial series. Given that the last term is ''ar''n and the previous series remainder is ''s'' - ''s''n-1 = ''ar''n / (1 - ''r'')), this measure of the convergence rate of the geometric series is ''ar''n / (''ar''n / (1 - ''r'')) = 1 - ''r'', if 0 ≤ ''r'' < 1. If ''r'' < 0, adjacent terms in the geometric series alternate between being positive and negative. A geometric interpretation of a converging alternating geometric series is shown in the adjacent diagram in which the areas of the negative terms are shown below the x axis. Pairing and summing each positive area with its negative smaller area neighbor results in non-overlapped trapezoids separated by gaps. To remove the gaps, broaden each trapezoid to cover the rightmost 1 - ''r''2 of the original triangle area instead of just the rightmost 1 - , ''r'', . However, to maintain the same trapezoid areas during this broadening transformation, scaling is needed: scale*(1 - ''r''2) = (1 - , ''r'', ), or scale = (1 - , ''r'', ) / (1 - ''r''2) = (1 + ''r'') / (1 - ''r''2) = (1 + ''r'') / ((1 + ''r'')(1 - ''r'')) = 1 / (1 - ''r'') where -1 < ''r'' ≤ 0. Note that because ''r'' < 0 this scale decreases the amplitude of the separated trapezoids in order to fill in the separation gaps. In contrast, for the case ''r'' > 0 the same scale 1 / (1 - ''r'') increases the amplitude of the non-overlapped trapezoids in order to account for the loss of the overlapped areas. With the gaps removed, pairs of terms in a converging alternating geometric series become a converging (non-alternating) geometric series with common ratio ''r''2 to account for the pairing of terms, coefficient ''a'' = 1 / (1 - ''r'') to account for the gap filling, and the degree (i.e., highest powered term) of the partial series called m instead of n to emphasize that terms have been paired. Similar to the ''r'' > 0 case, the ''r'' < 0 convergence rate = ''ar''2m / (''s'' - sm-1) = 1 - ''r''2, which is the same as the convergence rate of a non-alternating geometric series if its terms were similarly paired. Therefore, the convergence rate does not depend upon n or m and, perhaps more surprising, does not depend upon the sign of the common ratio. One perspective that helps explain the variable rate of convergence that is symmetric about ''r'' = 0 is that each added term of the partial series makes a finite contribution to the infinite sum at ''r'' = 1 and each added term of the partial series makes a finite contribution to the infinite slope at ''r'' = -1.

## Derivation

### Finite series

To derive this formula, first write a general geometric series as: $\sum_^ ar^ = ar^0+ar^1+ar^2+ar^3+\cdots+ar^.$ We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − ''r'', and we'll see that $\begin (1-r) \sum_^ ar^ & = (1-r)(ar^0 + ar^1+ar^2+ar^3+\cdots+ar^) \\ & = ar^0 + ar^1+ar^2+ar^3+\cdots+ar^ - ar^1-ar^2-ar^3-\cdots-ar^ - ar^n \\ & = a - ar^n \end$ since all the other terms cancel. If ''r'' ≠ 1, we can rearrange the above to get the convenient formula for a geometric series that computes the sum of n terms: $\sum_^ ar^ = \frac.$ ; Related formulas If one were to begin the sum not from k=1 or 0 but from a different value, say , then $\begin \sum_^n ar^ &= \begin \frac & \textr \neq 1 \\ a(n-m+1) & \textr = 1 \end\\ \sum_^n ar^k &= \begin a(n-m+1) & \textr = 1 \\ \frac & \textr \neq 1 \end\end$ Differentiating this formula with respect to allows us to arrive at formulae for sums of the form $G_s(n, r) := \sum_^n k^s r^k.$ For example: $\frac\sum_^nr^k = \sum_^n kr^= \frac-\frac.$ For a geometric series containing only even powers of multiply by : $\begin (1-r^2) \sum_^ ar^ &= a-ar^\\ \sum_^ ar^ &= \frac \end$ Equivalently, take as the common ratio and use the standard formulation. For a series with only odd powers of , $\begin (1-r^2) \sum_^ ar^ &= ar-ar^\\ \sum_^ ar^ &= \frac &= \frac \end$ An exact formula for the generalized sum $G_s\left(n, r\right)$ when $s \in \mathbb$ is expanded by the
Stirling numbers of the second kind In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
as

### Infinite series

An infinite geometric series is an
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
whose successive terms have a common ratio. Such a series converges
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
of the common ratio is less than one ( < 1). Its value can then be computed from the finite sum formula :$\sum_^\infty ar^k = \lim_ = \lim_\frac= \frac - \lim_$ Since: :$r^ \to 0 \mbox n \to \infty \mbox , r, < 1.$ Then: :$\sum_^\infty ar^k = \frac - 0 = \frac$ For a series containing only even powers of $r$, :$\sum_^\infty ar^ = \frac$ and for odd powers only, :$\sum_^\infty ar^ = \frac$ In cases where the sum does not start at ''k'' = 0, :$\sum_^\infty ar^k=\frac$ The formulae given above are valid only for  < 1. The latter formula is valid in every
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
, as long as the norm of ''r'' is less than one, and also in the field of ''p''-adic numbers if ''p'' < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums. For example, :$\frac\sum_^\infty r^k = \sum_^\infty kr^= \frac$ This formula only works for  < 1 as well. From this, it follows that, for  < 1, :$\sum_^ k r^k = \frac \,;\, \sum_^ k^2 r^k = \frac \, ; \, \sum_^ k^3 r^k = \frac$ Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is an elementary example of a series that converges absolutely. It is a
geometric series In mathematics, a geometric series 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 suc ...
whose first term is 1/2 and whose common ratio is 1/2, so its sum is :$\frac12+\frac14+\frac18+\frac+\cdots=\frac = 1.$ The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an
alternating series In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternati ...
that converges absolutely. It is a
geometric series In mathematics, a geometric series 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 suc ...
whose first term is 1/2 and whose common ratio is −1/2, so its sum is :$\frac12-\frac14+\frac18-\frac+\cdots=\frac = \frac13.$

### Complex series

The summation formula for geometric series remains valid even when the common ratio is a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
. In this case the condition that the absolute value of ''r'' be less than 1 becomes that the modulus of ''r'' be less than 1. It is possible to calculate the sums of some non-obvious geometric series. For example, consider the proposition : $\sum_^ \frac = \frac$ The proof of this comes from the fact that : $\sin\left(kx\right) = \frac ,$ which is a consequence of
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
. Substituting this into the original series gives :

# History

## Zeno of Elea (c.495 – c.430 BC)

2,500 years ago, Greek mathematicians had a problem when walking from one place to another: 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 Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known ...
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 inked surface resting upon a print medium (such as paper or cloth), thereby transferring the ink. It marked a dramatic improvement on earlier printing methods in which th ...
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 (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientist ...
used the sum of a geometric series to compute the area enclosed by a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
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: :$1 \,+\, 2\left\left(\frac\right\right) \,+\, 4\left\left(\frac\right\right)^2 \,+\, 8\left\left(\frac\right\right)^3 \,+\, \cdots.$ 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 of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area ...
, an early version of
integration Integration may refer to: Biology *Multisensory integration * Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technolog ...
. Using
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...
, the same area could be found by a
definite integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
.

## 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 of the later Middle Ages. He wrote influential works on economics, mathematics, physics, astrology a ...
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, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
and long division reveals the coefficients of these multi-dimensional geometric series, where the closed form is valid only within the range , ''r'', <1. :$\begin \text \\ \text \quad \text \\ \text \quad \text \quad \text \\ \text \quad\text \quad \text \quad\text \\ \text \quad\text \quad \text \quad \text \quad \text \end$ :$\begin &d \quad S^d / a\ \text \quad &&S^d / a\ \text \\ &1 \quad 1 / \left(1-r\right) \quad &&1 + r + r^2 + r^3 + r^4 + \cdots \\ &2 \quad 1 / \left(1-r\right)^2 \quad &&1 + 2r + 3r^2 + 4r^3 + 5r^4 + \cdots \\ &3 \quad 1 / \left(1-r\right)^3 \quad &&1 + 3r + 6r^2 + 10r^3 + 15r^4 + \cdots \\ &4 \quad 1 / \left(1-r\right)^4 \quad &&1 + 4r + 10r^2 + 20r^3 + 35r^4 + \cdots \\ \end$ As an aside, instead of using 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 discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in th ...
. 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.

# Applications

## Economics

In
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, geometric series are used to represent the
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
of an annuity (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 that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as con ... . Receiving$100 a year from now is worth less than an immediate $100, because one cannot invest Investment is the dedication of money to purchase 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, the purpose of investing is ... 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 (), in civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners to raise funds for any ...
). It can also be used to estimate the present value of expected stock dividends, or the terminal value of a
financial asset A financial asset is a non-physical asset whose value is derived from a contractual claim, such as bank deposits, bonds, and participations in companies' share capital. Financial assets are usually more liquid than other tangible assets, such a ...
assuming a stable growth rate.

## Fractal geometry

For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (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 :$1 \,+\, 3\left\left(\frac\right\right) \,+\, 12\left\left(\frac\right\right)^2 \,+\, 48\left\left(\frac\right\right)^3 \,+\, \cdots.$ 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.

## Integration

The derivative of $f\left(x\right) = \arctan\left(u\left(x\right)\right) \text f\text{'}\left(x\right) = u\text{'}\left(x\right)/\left(1+$
(x) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
2) because, letting $y \textu \text f\left(x\right) \text u\left(x\right),$ :$\begin y &= \arctan\left(u\right) &&\quad \text \\ u &= \tan\left(y\right) &&\quad \text -\pi/2 < y < \pi/2 \text \\ u\text{'} &= \sec^2y \cdot y\text{'} &&\quad \text \tan\left(y\right) = \sin\left(y\right)/\cos\left(y\right), \\ y\text{'} &= u\text{'}/\sec^2y &&\quad \text \sec^2y, \\ &= u\text{'}/\left(1+\tan^2y\right) &&\quad \text \sin^2y + \cos^2y = 1 \text \cos^2y, \\ &= u\text{'}/\left(1+u^2\right) &&\quad \text u = \tan\left(y\right). \end$ Therefore, letting $u\left(x\right) = x, \arctan\left(x\right)$ is the integral :$\begin \arctan\left(x\right)&=\int\frac \quad &&\text -\pi/2 < \arctan\left(x\right) < \pi/2,\\ &=\int\frac \quad &&\textr = -x^2,\\ &=\int\left\left(1 + \left\left(-x^2\right\right) + \left\left(-x^2\right\right)^2 + \left\left(-x^2\right\right)^3+\cdots\right\right)dx \quad &&\text,\\ &=\int\left\left(1-x^2+x^4-x^6+\cdots\right\right)dx \quad &&\text,\\ &=x-\frac+\frac-\frac+\cdots \quad &&\text,\\ &=\sum^_ \frac x^ \quad &&\text, \end$ which is called Gregory's series and is commonly attributed to Madhava of Sangamagrama (c. 1340 – c. 1425).

# Instances

* : 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 In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an 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 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 start the sequence from ...
(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, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
is when the common ratio ''r'' equals the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ...
(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 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 start the sequence from ...
has the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ...
as its coefficient ''a'' and the conjugate
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ...
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 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 start the sequence from ...
because 1 + ''r'' = ''r''2, and it is an
alternating series In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternati ...
because ''r'' < 0.

## Geometric series

The geometric series has two degrees of freedom: one for its coefficient ''a'' and another for its common ratio ''r''. In the map of polynomials, the big red circle represents the set of all geometric series.

### Converging geometric series

Only a subset of all geometric series converge. Specifically, a geometric series converges if and only if its common ratio , ''r'', < 1. In the map of polynomials, the red triangle represents the set of converging geometric series and being drawn inside the big red circle representing the set of all geometric series indicates the converging geometric series is a subset of the geometric series.

### = Repeated decimals

= Only a subset of all converging geometric series converge to decimal fractions that have repeated patterns that continue forever (e.g., 0.7777... or 0.9999... or 0.123412341234...). In the map of polynomials, the little yellow triangle represents the set of geometric series that converge to infinitely repeated decimal patterns. It is drawn inside the red triangle to indicate it is a subset of the converging geometric series, which in turn is drawn inside the big red circle indicating both the converging geometric series and the geometric series that converge to infinitely repeated patterns are subsets of the geometric series. Although fractions with infinitely repeated decimal patterns can only be approximated when encoded as floating point numbers, they can always be defined exactly as the ratio of two integers and those two integers can be calculated using the geometric series. For example, the repeated decimal fraction 0.7777... can be written as the geometric series :$0.7777\ldots \;=\; \frac \,+\, \frac\frac \,+\, \frac\frac \,+\, \frac\frac \,+\, \cdots$ where coefficient ''a'' = 7/10 and common ratio ''r'' = 1/10. The geometric series closed form reveals the two integers that specify the repeated pattern: :$0.7777\ldots \;=\; \frac \;=\; \frac \;=\; \frac \;=\; \frac.$ This approach extends beyond base-ten numbers. In fact, any fraction that has an infinitely repeated pattern in base-ten numbers also has an infinitely repeated pattern in numbers written in any other base. For example, looking at the floating point encoding for the number 0.7777...  julia> bitstring(Float32(0.77777777777777777777)) "00111111010001110001110001110010"  reveals the binary fraction 0.110001110001110001... where the binary pattern 0b110001 repeats indefinitely and can be written in mostly (except for the powers) binary numbers as :$0.110001110001110001\ldots \;=\; \frac \,+\, \frac\frac \,+\, \frac\frac \,+\, \frac\frac \,+\, \cdots$ where coefficient ''a'' = 0b110001 / 0b1000000 = 49 / 64 and common ratio ''r'' = 1 / 0b1000000 = 1 / 64. Using the geometric series closed form as before :$0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac \;=\; \frac \;=\; \frac \;=\; \frac \;=\; \frac.$ You may have noticed that the floating point encoding does not capture the 0b110001 repeat pattern in the last couple (least significant) bits. This is because floating point encoding rounds the remainder instead of truncating it. Therefore, if the most significant bit of the remainder is 1, the least significant bit of the encoded fraction gets incremented and that will cause a carry if the least significant bit of the fraction is already 1, which can cause another carry if that bit of the fraction is already a 1, which can cause another carry, etc. This floating point rounding and the subsequent carry propagation explains why the floating point encoding for 0.99999... is exactly the same as the floating point encoding for 1.  julia> bitstring(Float32(0.99999999999999999999)) "00111111100000000000000000000000" julia> bitstring(Float32(1.0)) "00111111100000000000000000000000"  As an example that has four digits in the repeated pattern, 0.123412341234... can be written as the geometric series :$0.123412341234\ldots \;=\; \frac \,+\, \frac\frac \,+\, \frac\frac \,+\, \frac\frac \,+\, \cdots$ where coefficient ''a'' = 1234/10000 and common ratio ''r'' = 1/10000. The geometric series closed form reveals the two integers that specify the repeated pattern: :$0.123412341234\ldots \;=\; \frac \;=\; \frac \;=\; \frac \;=\; \frac.$

## Power series

Like the geometric series, the power series has one degree of freedom for its common ratio ''r'' (along the x-axis) but has ''n''+1 degrees of freedom for its coefficients (along the y-axis), where ''n'' represents the power of the last term in the partial series. In the map of polynomials, the big blue circle represents the set of all power series.

### = Binary encoded numbers

= Zeno of Elea's geometric series with coefficient ''a''=1/2 and common ratio ''r''=1/2 is the foundation of binary encoded approximations of
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
in digital computers. Concretely, the geometric series written in its normalized vector form is ''s''/''a'' = 1 1 1 1 …1 ''r'' ''r''2 ''r''3 ''r''4 …]T. Keeping the column vector of basis functions ''r'' ''r''2 ''r''3 ''r''4sup>T the same but generalizing the row vector 1 1 1 1 …so that each entry can be either a 0 or a 1 allows for an approximate encoding of any fraction. For example, the value ''v'' = 0.34375 is encoded as  ''v''/''a'' = 1 0 1 1 0 … ''r'' ''r''2 ''r''3 ''r''4sup>T where coefficient ''a'' = 1/2 and common ratio ''r'' = 1/2. Typically, the row vector is written in the more compact binary form ''v'' = 0.010110 which is 0.34375 in decimal. Similarly, the geometric series with coefficient ''a''=1 and common ratio ''r''=2 is the foundation for binary encoded integers in digital computers. Again, the geometric series written in its normalized vector form is ''s''/''a'' = 1 1 1 1 …1 ''r'' ''r''2 ''r''3 ''r''4 …]T. Keeping the column vector of basis functions ''r'' ''r''2 ''r''3 ''r''4sup>T the same but generalizing the row vector 1 1 1 1 …so that each entry can be either a 0 or a 1 allows for an encoding of any integer. For example, the value ''v'' = 151 is encoded as  ''v''/''a'' = 1 1 0 1 0 0 1 0 …1 ''r'' ''r''2 ''r''3 ''r''4 ''r''5 ''r''6 ''r''7 ''r''8 …]T where coefficient ''a'' = 1 and common ratio ''r'' = 2. Typically, the row vector is written in reverse order (so that the most significant bit is first) in the more compact binary form ''v'' = …010010111 = 10010111 which is 151 in decimal. As shown in the adjacent figure, the standard binary encoding of a 32-bit floating point number is a combination of a binary encoded integer and a binary encoded fraction, beginning at the most significant bit with :* the sign bit, followed by :* an 8-bit integer exponent field with an assumed offset of 127 (so a value of 127 represents an exponent value of 0) and with a base of 2 meaning that the exponent value specifies a bit shift of the fraction field, followed by :* a 23-bit fraction field with an assumed but not encoded 1 serving as the fraction's most significant nonzero bit which would be in bit position 23 if it were encoded. Building upon the previous example of 0.34375 having binary encoding of 0.010110, a floating point encoding (according to the IEEE 754 standard) of 0.34375 is :* the sign bit which is 0 because the number is not negative, :* an 8-bit integer exponent field which must specify a shift that counters the 2 bit left shift to get the original binary encoding from 0.010110 to 1.0110, and that counter shift to recover the original binary encoding is a right shift of 2 bits which is specified by an exponent value of 125 (because 125 − 127 = -2 which is a right shift of 2 bits) which in binary is 0111 1101, :* a 23-bit fraction field: .0110 0000 0000 0000 0000 000. Although encoding floating point numbers by hand like this is possible, letting a computer do it is easier and less error prone. The following Julia code confirms the hand calculated floating point encoding of the number 0.34375:  julia> bitstring(Float32(0.34375)) "00111110101100000000000000000000" 

## Laurent series

### Complex Fourier series

As an example of the ability of the complex Fourier series to trace any 2D closed figure, in the adjacent animation a complex Fourier series traces the letter 'e' (for exponential). Given the intricate coordination of motions shown in the animation, a definition of the complex Fourier series can be surprisingly compact in just two equations: :$\begin s\left(t\right) &= \sum_^\infty c_n e^ \\ c_n &= \int_^1 s\left(t\right) e^ dt \\ \end$ where parameterized function ''s''(''t'') traces some 2D closed figure in the complex plane as the parameter ''t'' progresses through the period from 0 to 1. To help make sense of these compact equations defining the complex Fourier series, note that the complex Fourier series summation looks similar to the complex geometric series except that the complex Fourier series is basically two complex geometric series (one set of terms rotating in the positive direction and another set of terms rotating in the negative direction), and the coefficients of the complex Fourier series are complex constants that can vary from term to term. By allowing terms to rotate in either direction, the series becomes capable of tracing any 2D closed figure. In contrast, the complex geometric series has all the terms rotating in the same direction and it can trace only circles. Allowing the coefficients of the complex geometric series to vary from term to term would expand upon the shapes it can trace but all the possible shapes would still be limited to being puffy and cloud-like, not able to trace the shape of a simple line segment, for example going back and forth between 1 + i0 and -1 + i0. However, Euler's formula shows that the addition of just two terms rotating in opposite directions can trace that line segment between 1 + i0 and -1 + i0: :$\begin e^ &= \cos\theta + i\sin\theta \\ e^ &= \cos\theta - i\sin\theta \\ \cos\theta &= \frac. \\ \end$ Concerning the complex Fourier series second equation defining how to calculate the coefficients, the coefficient of the non-rotating term ''c''0 can be calculated by integrating the complex Fourier series first equation over the range of one period from 0 to 1. Over that range, all the rotating terms integrate to zero, leaving just ''c''0. Similarly, any of the terms in the complex Fourier series first equation can be made to be a non-rotating term by multiplying both sides of the equation by $e^$ before integrating to calculate ''c''n, and that is the complex Fourier series second equation.

## Matrix polynomial

### Matrix exponential

* * * * * * * * * *

# 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. * * Theoni Pappas, 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 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. He teaches the history of mathematics at Loyola Uni ...
(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.