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
:
is geometric, because each successive term can be obtained by multiplying the previous term by
. In general, a geometric series is written as
, where
is the
coefficient of each term and
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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the
Taylor series, 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.
The name geometric series indicates each term is the
geometric mean 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 co ...
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 called ...
of geometric
series 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 ...
'' 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 ...
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
:
''ar''
k
and a
closed form of the geometric series written as
:
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, 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: 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, 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,
:* from some of the largest observations such as the
expansion of the universe 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'', e
i''θ'' where , ''r'', is the
vector's magnitude (or length), ''θ'' is the vector's angle (or orientation) in the
complex plane and i
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'',
2e
i2''θ'' + ''a'', ''r'',
3e
i3''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'', e
i''ω''0''t'' + ''a'', ''r'',
2e
i2''ω''0''t'' + ''a'', ''r'',
3e
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
harmonics of the fundamental angular frequency ''ω''
0. The constraint , ''r'', <1 is enough to coordinate this infinite number of vectors of different lengths all rotating at different speeds into tracing a circle, as shown in the adjacent video. Similar to how the
Taylor series 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:
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 terms as follows:
As approaches infinity, the absolute value of must be less than one for the series to converge. The sum then becomes
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. That is the most common interpretation of the distance between two values. However, the
p-adic 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 mathe ...
, 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, "Ma ...
, 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:
:
The second equality is true because if
then
as
and
:
Alternatively, a geometric interpretation of the convergence is shown in the adjacent diagram. The area of the white triangle is the series
remainder = ''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'' - 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.
Derivation
Finite series
To derive this formula, first write a general geometric series as:
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
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:
; Related formulas
If one were to begin the sum not from k=1 or 0 but from a different value, say , then
Differentiating this formula with respect to allows us to arrive at formulae for sums of the form
For example:
For a geometric series containing only even powers of multiply by :
Equivalently, take as the common ratio and use the standard formulation.
For a series with only odd powers of ,
An exact formula for the generalized sum
when
is expanded by the
Stirling numbers of the second kind as
Infinite series
An infinite geometric series is an
infinite series 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 bic ...
the
absolute value of the common ratio is less than one ( < 1). Its value can then be computed from the finite sum formula
:
Since:
:
Then:
:
For a series containing only even powers of
,
:
and for odd powers only,
:
In cases where the sum does not start at ''k'' = 0,
:
The formulae given above are valid only for < 1. The latter formula is valid in every
Banach algebra, 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,
:
This formula only works for < 1 as well. From this, it follows that, for < 1,
:
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 whose first term is 1/2 and whose common ratio is 1/2, so its sum is
:
The inverse of the above series is
1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an
alternating series that converges absolutely.
It is a
geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is
:
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
:
The proof of this comes from the fact that
:
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 ...
. Substituting this into the original series gives
: