In

^{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 _{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 ^{k}
and a closed form of the geometric series written as
:$\backslash frac\; \backslash text\; ,\; r,\; <1.$
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

^{2} + ''ar''^{3} + ... is an infinite series defined by just two ^{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 ^{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 ^{i''θ''} where , ''r'', is the ^{2} = -1. With a common ratio , ''r'', e^{i''θ''}, the expanded form of the geometric series is ''a'' + ''a'', ''r'', e^{i''θ''} + ''a'', ''r'', ^{2}e^{i2''θ''} + ''a'', ''r'', ^{3}e^{i3''θ''} + ... Modeling the angle ''θ'' as linearly increasing over time at the rate of some _{0} (in other words, making the substitution ''θ'' = ''ω''_{0}''t''), the expanded form of the geometric series becomes ''a'' + ''a'', ''r'', e^{i''ω''0''t''} + ''a'', ''r'', ^{2}e^{i2''ω''0''t''} + ''a'', ''r'', ^{3}e^{i3''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 _{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

^{n-1} term, is given by the closed-form formula:
$$\backslash begin\; s\_n\; \&=\; ar^0\; +\; ar^1\; +\; \backslash cdots\; +\; ar^\backslash \backslash \; \&=\; \backslash sum\_^\; ar^k\; =\; \backslash sum\_^\; ar^\backslash \backslash \; \&=\; \backslash begin\; a\backslash left(\backslash frac\backslash right),\; \backslash text\; r\; \backslash neq\; 1\backslash \backslash \; an,\; \backslash text\; r\; =\; 1\; \backslash end\; \backslash end$$
where is the common ratio. One can derive that closed-form formula for the partial sum, ''s''_{n}, by subtracting out the many

_{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.

^{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'' - s_{m-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.

_{''p''} < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums.
For example,
:$\backslash frac\backslash sum\_^\backslash infty\; r^k\; =\; \backslash sum\_^\backslash infty\; kr^=\; \backslash frac$
This formula only works for < 1 as well. From this, it follows that, for < 1,
:$\backslash sum\_^\; k\; r^k\; =\; \backslash frac\; \backslash ,;\backslash ,\; \backslash sum\_^\; k^2\; r^k\; =\; \backslash frac\; \backslash ,\; ;\; \backslash ,\; \backslash sum\_^\; k^3\; r^k\; =\; \backslash frac$
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

^{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,
^{2}, ar^{3} 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:

_{''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.

^{2} (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is
:$\backslash sum\_^\backslash infty\; \backslash frac,$
which is the infinite series:
:$\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash cdots.$
This is a geometric series with common ratio 1 / (1 + $I$ ). The sum is the first term divided by (one minus the common ratio):
:$\backslash frac\; \backslash ;=\backslash ;\; \backslash frac.$
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

_{n} = F_{n-1} + F_{n-2} but without requiring F_{0} = 0 and F_{1} = 1) when a geometric series common ratio ''r'' satisfies the constraint 1 + ''r'' = ''r''^{2}, which according to the ^{2}, and it is an

^{2} ''r''^{3} ''r''^{4} …]^{T}. Keeping the column vector of basis functions 2 ''r''3 ''r''4 …"> ''r'' ''r''^{2} ''r''^{3} ''r''^{4} …sup>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 …2 ''r''3 ''r''4 …"> ''r'' ''r''^{2} ''r''^{3} ''r''^{4} …sup>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 2 ''r''3 ''r''4 …"> ''r'' ''r''^{2} ''r''^{3} ''r''^{4} …sup>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:

_{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.

"Geometric Series"

by Michael Schreiber,

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
:$\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash cdots$
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''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''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
:$\backslash sum^\_$ ''ar''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''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''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''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'', evector
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 iangular 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''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''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''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:
$$\backslash begin\; s\_n\; \&=\; ar^0\; +\; ar^1\; +\; \backslash cdots\; +\; ar^,\backslash \backslash \; rs\_n\; \&=\; ar^1\; +\; ar^2\; +\; \backslash cdots\; +\; ar^,\backslash \backslash \; s\_n\; -\; rs\_n\; \&=\; ar^0\; -\; ar^,\backslash \backslash \; s\_n\backslash left(1-r\backslash right)\; \&=\; a\backslash left(1-r^\backslash right),\backslash \backslash \; s\_n\; \&=\; a\backslash left(\backslash frac\backslash right),\; \backslash text\; r\; \backslash neq\; 1.\; \backslash end$$
As approaches infinity, the absolute value of must be less than one for the series to converge. The sum then becomes
$$\backslash begin\; s\; \&=\; a+ar+ar^2+ar^3+ar^4+\backslash cdots\backslash \backslash \; \&=\; \backslash sum\_^\backslash infty\; ar^\; =\; \backslash sum\_^\backslash infty\; ar^\backslash \backslash \; \&=\; \backslash frac,\; \backslash text\; ,\; r,\; <1.\; \backslash end$$
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 ageometric 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 ...

:
:$\backslash begin\; 1\; +\; r\; +\; r^2\; +\; r^3\; +\; \backslash cdots\; \backslash \; \&=\; \backslash lim\_\; \backslash left(1\; +\; r\; +\; r^2\; +\; \backslash cdots\; +\; r^n\backslash right)\; \backslash \backslash \; \&=\; \backslash lim\_\; \backslash frac.\; \backslash end$
The second equality is true because if $,\; r,\; <\; 1,$ then $r^\; \backslash to\; 0$ as $n\; \backslash to\; \backslash infty$ and
:$\backslash begin\; (1\; +\; r\; +\; r^2\; +\; \backslash cdots\; +\; r^n)(1\; -\; r)\; \&=\; ((1-r)\; +\; (r\; -\; r^2)\; +\; (r^2\; -\; r^3)\; +\; ...\; +\; (r^n\; -\; r^))\backslash \backslash \; \&=\; (1\; +\; (-r\; +\; r)\; +\; (\; -r^2\; +\; r^2)\; +\; ...\; +\; (-r^n\; +\; r^n)\; -\; r^)\backslash \backslash \; \&=\; 1-r^.\; \backslash end$
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''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''Derivation

Finite series

To derive this formula, first write a general geometric series as: $$\backslash sum\_^\; ar^\; =\; ar^0+ar^1+ar^2+ar^3+\backslash cdots+ar^.$$ 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 $$\backslash begin\; (1-r)\; \backslash sum\_^\; ar^\; \&\; =\; (1-r)(ar^0\; +\; ar^1+ar^2+ar^3+\backslash cdots+ar^)\; \backslash \backslash \; \&\; =\; ar^0\; +\; ar^1+ar^2+ar^3+\backslash cdots+ar^\; -\; ar^1-ar^2-ar^3-\backslash cdots-ar^\; -\; ar^n\; \backslash \backslash \; \&\; =\; a\; -\; ar^n\; \backslash end$$ 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: $$\backslash sum\_^\; ar^\; =\; \backslash frac.$$ ; Related formulas If one were to begin the sum not from k=1 or 0 but from a different value, say , then $$\backslash begin\; \backslash sum\_^n\; ar^\; \&=\; \backslash begin\; \backslash frac\; \&\; \backslash textr\; \backslash neq\; 1\; \backslash \backslash \; a(n-m+1)\; \&\; \backslash textr\; =\; 1\; \backslash end\backslash \backslash \; \backslash sum\_^n\; ar^k\; \&=\; \backslash begin\; a(n-m+1)\; \&\; \backslash textr\; =\; 1\; \backslash \backslash \; \backslash frac\; \&\; \backslash textr\; \backslash neq\; 1\; \backslash end\backslash end$$ Differentiating this formula with respect to allows us to arrive at formulae for sums of the form $$G\_s(n,\; r)\; :=\; \backslash sum\_^n\; k^s\; r^k.$$ For example: $$\backslash frac\backslash sum\_^nr^k\; =\; \backslash sum\_^n\; kr^=\; \backslash frac-\backslash frac.$$ For a geometric series containing only even powers of multiply by : $$\backslash begin\; (1-r^2)\; \backslash sum\_^\; ar^\; \&=\; a-ar^\backslash \backslash \; \backslash sum\_^\; ar^\; \&=\; \backslash frac\; \backslash end$$ Equivalently, take as the common ratio and use the standard formulation. For a series with only odd powers of , $$\backslash begin\; (1-r^2)\; \backslash sum\_^\; ar^\; \&=\; ar-ar^\backslash \backslash \; \backslash sum\_^\; ar^\; \&=\; \backslash frac\; \&=\; \backslash frac\; \backslash end$$ An exact formula for the generalized sum $G\_s(n,\; r)$ when $s\; \backslash in\; \backslash mathbb$ is expanded by theStirling 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
$$G\_s(n,\; r)\; =\; \backslash sum\_^s\; \backslash left\backslash lbrace\backslash right\backslash rbrace\; x^j\; \backslash frac\backslash left;\; href="/html/ALL/s/frac\backslash right.html"\; ;"title="frac\backslash right">frac\backslash right$$
Infinite series

An infinite geometric series is aninfinite 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
:$\backslash sum\_^\backslash infty\; ar^k\; =\; \backslash lim\_\; =\; \backslash lim\_\backslash frac=\; \backslash frac\; -\; \backslash lim\_$
Since:
:$r^\; \backslash to\; 0\; \backslash mbox\; n\; \backslash to\; \backslash infty\; \backslash mbox\; ,\; r,\; <\; 1.$
Then:
:$\backslash sum\_^\backslash infty\; ar^k\; =\; \backslash frac\; -\; 0\; =\; \backslash frac$
For a series containing only even powers of $r$,
:$\backslash sum\_^\backslash infty\; ar^\; =\; \backslash frac$
and for odd powers only,
:$\backslash sum\_^\backslash infty\; ar^\; =\; \backslash frac$
In cases where the sum does not start at ''k'' = 0,
:$\backslash sum\_^\backslash infty\; ar^k=\backslash frac$
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 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
:$\backslash frac12+\backslash frac14+\backslash frac18+\backslash frac+\backslash cdots=\backslash frac\; =\; 1.$
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
:$\backslash frac12-\backslash frac14+\backslash frac18-\backslash frac+\backslash cdots=\backslash frac\; =\; \backslash frac13.$
Complex series

The summation formula for geometric series remains valid even when the common ratio is acomplex 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
: $\backslash sum\_^\; \backslash frac\; =\; \backslash frac$
The proof of this comes from the fact that
: $\backslash sin(kx)\; =\; \backslash frac\; ,$
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
: $\backslash sum\_^\; \backslash frac\; =\; \backslash frac\; \backslash left;\; href="/html/ALL/s/\backslash sum\_^\_\backslash left(\_\backslash frac\_\backslash right)^k\_-\_\backslash sum\_^\_\backslash left(\backslash frac\backslash right)^k\backslash right.html"\; ;"title="\backslash sum\_^\; \backslash left(\; \backslash frac\; \backslash right)^k\; -\; \backslash sum\_^\; \backslash left(\backslash frac\backslash right)^k\backslash right">\backslash sum\_^\; \backslash left(\; \backslash frac\; \backslash right)^k\; -\; \backslash sum\_^\; \backslash left(\backslash frac\backslash right)^k\backslash right$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 whenZeno 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 theprinting 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''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, ar

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:
:$1\; \backslash ,+\backslash ,\; 2\backslash left(\backslash frac\backslash right)\; \backslash ,+\backslash ,\; 4\backslash left(\backslash frac\backslash right)^2\; \backslash ,+\backslash ,\; 8\backslash left(\backslash frac\backslash right)^3\; \backslash ,+\backslash ,\; \backslash cdots.$
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
:$1\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\; \backslash ,+\backslash ,\; \backslash cdots.$
This is a geometric series with common ratio and the fractional part is equal to
:$\backslash sum\_^\backslash infty\; 4^\; =\; 1\; +\; 4^\; +\; 4^\; +\; 4^\; +\; \backslash cdots\; =\; .$
The sum is
:$\backslash frac\backslash ;=\backslash ;\backslash frac\backslash ;=\backslash ;\backslash frac.$
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''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.
:$\backslash begin\; \backslash text\; \backslash \backslash \; \backslash text\; \backslash quad\; \backslash text\; \backslash \backslash \; \backslash text\; \backslash quad\; \backslash text\; \backslash quad\; \backslash text\; \backslash \backslash \; \backslash text\; \backslash quad\backslash text\; \backslash quad\; \backslash text\; \backslash quad\backslash text\; \backslash \backslash \; \backslash text\; \backslash quad\backslash text\; \backslash quad\; \backslash text\; \backslash quad\; \backslash text\; \backslash quad\; \backslash text\; \backslash end$
:$\backslash begin\; \&d\; \backslash quad\; S^d\; /\; a\backslash \; \backslash text\; \backslash quad\; \&\&S^d\; /\; a\backslash \; \backslash text\; \backslash \backslash \; \&1\; \backslash quad\; 1\; /\; (1-r)\; \backslash quad\; \&\&1\; +\; r\; +\; r^2\; +\; r^3\; +\; r^4\; +\; \backslash cdots\; \backslash \backslash \; \&2\; \backslash quad\; 1\; /\; (1-r)^2\; \backslash quad\; \&\&1\; +\; 2r\; +\; 3r^2\; +\; 4r^3\; +\; 5r^4\; +\; \backslash cdots\; \backslash \backslash \; \&3\; \backslash quad\; 1\; /\; (1-r)^3\; \backslash quad\; \&\&1\; +\; 3r\; +\; 6r^2\; +\; 10r^3\; +\; 15r^4\; +\; \backslash cdots\; \backslash \backslash \; \&4\; \backslash quad\; 1\; /\; (1-r)^4\; \backslash quad\; \&\&1\; +\; 4r\; +\; 10r^2\; +\; 20r^3\; +\; 35r^4\; +\; \backslash cdots\; \backslash \backslash \; \backslash end$
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

Ineconomics
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 + $I$)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 :$1\; \backslash ,+\backslash ,\; 3\backslash left(\backslash frac\backslash right)\; \backslash ,+\backslash ,\; 12\backslash left(\backslash frac\backslash right)^2\; \backslash ,+\backslash ,\; 48\backslash left(\backslash frac\backslash right)^3\; \backslash ,+\backslash ,\; \backslash cdots.$ 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 :$1\backslash ,+\backslash ,\backslash frac\backslash ;=\backslash ;1\backslash ,+\backslash ,\backslash frac\backslash ;=\backslash ;\backslash frac.$ Thus the Koch snowflake has 8/5 of the area of the base triangle.Integration

The derivative of $f(x)\; =\; \backslash arctan(u(x))\; \backslash text\; f\text{'}(x)\; =\; u\text{'}(x)/(1+$(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 $y\; \backslash textu\; \backslash text\; f(x)\; \backslash text\; u(x),$
:$\backslash begin\; y\; \&=\; \backslash arctan(u)\; \&\&\backslash quad\; \backslash text\; \backslash \backslash \; u\; \&=\; \backslash tan(y)\; \&\&\backslash quad\; \backslash text\; -\backslash pi/2\; <\; y\; <\; \backslash pi/2\; \backslash text\; \backslash \backslash \; u\text{'}\; \&=\; \backslash sec^2y\; \backslash cdot\; y\text{'}\; \&\&\backslash quad\; \backslash text\; \backslash tan(y)\; =\; \backslash sin(y)/\backslash cos(y),\; \backslash \backslash \; y\text{'}\; \&=\; u\text{'}/\backslash sec^2y\; \&\&\backslash quad\; \backslash text\; \backslash sec^2y,\; \backslash \backslash \; \&=\; u\text{'}/(1+\backslash tan^2y)\; \&\&\backslash quad\; \backslash text\; \backslash sin^2y\; +\; \backslash cos^2y\; =\; 1\; \backslash text\; \backslash cos^2y,\; \backslash \backslash \; \&=\; u\text{'}/(1+u^2)\; \&\&\backslash quad\; \backslash text\; u\; =\; \backslash tan(y).\; \backslash end$
Therefore, letting $u(x)\; =\; x,\; \backslash arctan(x)$ is the integral
:$\backslash begin\; \backslash arctan(x)\&=\backslash int\backslash frac\; \backslash quad\; \&\&\backslash text\; -\backslash pi/2\; <\; \backslash arctan(x)\; <\; \backslash pi/2,\backslash \backslash \; \&=\backslash int\backslash frac\; \backslash quad\; \&\&\backslash textr\; =\; -x^2,\backslash \backslash \; \&=\backslash int\backslash left(1\; +\; \backslash left(-x^2\backslash right)\; +\; \backslash left(-x^2\backslash right)^2\; +\; \backslash left(-x^2\backslash right)^3+\backslash cdots\backslash right)dx\; \backslash quad\; \&\&\backslash text,\backslash \backslash \; \&=\backslash int\backslash left(1-x^2+x^4-x^6+\backslash cdots\backslash right)dx\; \backslash quad\; \&\&\backslash text,\backslash \backslash \; \&=x-\backslash frac+\backslash frac-\backslash frac+\backslash cdots\; \backslash quad\; \&\&\backslash text,\backslash \backslash \; \&=\backslash sum^\_\; \backslash frac\; x^\; \backslash quad\; \&\&\backslash text,\; \backslash end$
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, analternating 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 ...

(Fquadratic 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''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 :$0.7777\backslash ldots\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash cdots$ 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: :$0.7777\backslash ldots\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac.$ 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
:$0.110001110001110001\backslash ldots\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash cdots$
where coefficient ''a'' = 0b110001 / 0b1000000 = 49 / 64 and common ratio ''r'' = 1 / 0b1000000 = 1 / 64. Using the geometric series closed form as before
:$0.7777\backslash ldots\; \backslash ;=\backslash ;\; 0b0.110001110001110001\backslash ldots\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac.$
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
:$0.123412341234\backslash ldots\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash frac\backslash frac\; \backslash ,+\backslash ,\; \backslash cdots$
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:
:$0.123412341234\backslash ldots\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac\; \backslash ;=\backslash ;\; \backslash frac.$
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.Taylor 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 offractions
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''```
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: :$\backslash begin\; s(t)\; \&=\; \backslash sum\_^\backslash infty\; c\_n\; e^\; \backslash \backslash \; c\_n\; \&=\; \backslash int\_^1\; s(t)\; e^\; dt\; \backslash \backslash \; \backslash end$ 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: :$\backslash begin\; e^\; \&=\; \backslash cos\backslash theta\; +\; i\backslash sin\backslash theta\; \backslash \backslash \; e^\; \&=\; \backslash cos\backslash theta\; -\; i\backslash sin\backslash theta\; \backslash \backslash \; \backslash cos\backslash theta\; \&=\; \backslash frac.\; \backslash \backslash \; \backslash end$ Concerning the complex Fourier series second equation defining how to calculate the coefficients, the coefficient of the non-rotating term ''c''Matrix polynomial

Matrix exponential

See also

* * * * * * * * * *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. * * 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.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 hos ...

, 2007.
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